TOPICS
SURROUNDING
THE
COMBINATORIAL
ANABELIAN
GEOMETRY
OF
HYPERBOLIC
CURVES
IV:
DISCRETENESS
AND
SECTIONS
YUICHIRO
HOSHI
AND
SHINICHI
MOCHIZUKI
JULY
2022
Abstract.
Let
Σ
be
a
nonempty
subset
of
the
set
of
prime
numbers
which
is
either
equal
to
the
entire
set
of
prime
numbers
or
of
cardi-
nality
one.
In
the
present
paper,
we
continue
our
study
of
the
pro-Σ
fundamental
groups
of
hyperbolic
curves
and
their
associated
config-
uration
spaces
over
algebraically
closed
fields
in
which
the
primes
of
Σ
are
invertible.
The
present
paper
focuses
on
the
topic
of
compar-
ison
between
the
theory
developed
in
earlier
papers
concerning
pro-
Σ
fundamental
groups
and
various
discrete
versions
of
this
theory.
We
begin
by
developing
a
theory
of
combinatorial
analogues
of
the
section
conjecture
and
Grothendieck
conjecture
in
anabelian
geometry
for
abstract
combinatorial
versions
of
the
data
that
arises
from
a
hyperbolic
curve
over
a
complete
discretely
valued
field,
under
the
condition
that,
for
some
l
∈
Σ,
the
l-adic
cyclotomic
character
has
infinite
image.
This
portion
of
the
theory
is
purely
combina-
torial
and
essentially
follows
from
a
result
concerning
the
existence
of
fixed
points
of
actions
of
finite
groups
on
finite
graphs
[satisfying
certain
conditions]
—
a
result
which
may
be
regarded
as
a
geomet-
ric
interpretation
of
the
well-known
elementary
fact
that
free
pro-Σ
groups
are
torsion-free.
We
then
examine
various
applications
of
this
purely
combinatorial
theory
to
scheme
theory.
Next,
we
verify
various
results
in
the
theory
of
discrete
fundamental
groups
of
hy-
perbolic
topological
surfaces
to
the
effect
that
various
properties
of
[discrete]
subgroups
of
such
groups
hold
if
and
only
if
analogous
properties
hold
for
the
closures
of
these
subgroups
in
the
profinite
completions
of
the
discrete
fundamental
groups
under
considera-
tion.
These
results
make
possible
a
fairly
straightforward
trans-
lation,
into
discrete
versions,
of
pro-Σ
results
obtained
in
previous
papers
by
the
authors
concerning
the
theory
of
partial
combinatorial
cuspidalization,
Dehn
multi-twists,
the
tripod
homomorphism,
metric-
admissibility,
and
the
characterization
of
local
Galois
groups
in
the
global
Galois
image
associated
to
a
hyperbolic
curve.
Finally,
we
con-
sider
the
analogue
of
the
theory
of
tripods
[i.e.,
copies
of
the
pro-
Σ
or
discrete
fundamental
group
of
the
projective
line
minus
three
points]
associated
to
cycles
in
a
hyperbolic
topological
surface.
From
an
intuitive
topological
point
of
view,
these
tripods
are
obtained
by
2010
Mathematics
Subject
Classification.
Primary
14H30;
Secondary
14H10.
Key
words
and
phrases.
anabelian
geometry,
combinatorial
anabelian
geometry,
combinatorial
section
conjecture,
fixed
points,
combinatorial
Grothendieck
conjec-
ture,
combinatorial
cuspidalization,
discrete/profinite
comparison,
liftings
of
cycles,
tripods,
semi-graph
of
anabelioids,
semi-graph
of
temperoids,
hyperbolic
curve,
con-
figuration
space.
1
2
YUICHIRO
HOSHI
AND
SHINICHI
MOCHIZUKI
considering
once-punctured
tubular
neighborhoods
of
the
cy-
cles.
Such
a
construction
was
considered
previously
by
M.
Boggi
in
the
discrete
case,
but
in
the
present
paper,
we
consider
it
from
the
point
of
view
of
the
abstract
pro-Σ
theory
developed
in
earlier
pa-
pers
by
the
authors
and
then
proceed
to
relate
this
theory
to
the
discrete
theory
by
applying
the
tools
developed
in
earlier
portions
of
the
present
paper.
Contents
Introduction
2
0.
Notations
and
Conventions
12
1.
The
combinatorial
section
conjecture
13
2.
Discrete
combinatorial
anabelian
geometry
46
3.
Canonical
liftings
of
cycles
94
Appendix.
Explicit
limit
seminorms
associated
to
sequences
of
toric
surfaces
120
References
130
Introduction
Let
Σ
⊆
Primes
be
a
subset
of
the
set
of
prime
numbers
Primes
which
is
either
equal
to
Primes
or
of
cardinality
one.
In
the
present
paper,
we
continue
our
study
of
the
pro-Σ
fundamental
groups
of
hyperbolic
curves
and
their
associated
configuration
spaces
over
al-
gebraically
closed
fields
in
which
the
primes
of
Σ
are
invertible
[cf.
[CmbGC],
[MT],
[CmbCsp],
[NodNon],
[CbTpI],
[CbTpII],
[CbTpIII]].
The
present
paper
focuses
on
the
topic
of
understanding
the
relation-
ship
between
the
theory
developed
in
earlier
papers
concerning
pro-Σ
fundamental
groups
and
various
discrete
versions
of
this
theory.
This
topic
of
comparison
of
pro-Σ
and
discrete
versions
of
the
theory
turns
out
to
be
closely
related,
in
many
situations,
to
the
theory
of
sections
of
various
natural
surjections
of
profinite
groups.
Indeed,
this
rela-
tionship
with
the
theory
of
sections
is,
in
some
sense,
not
surprising,
inasmuch
as
sections
typically
amount
to
some
sort
of
fixed
point
within
a
profinite
continuum.
That
is
to
say,
such
fixed
points
are
often
closely
related
to
the
identification
of
a
rigid
discrete
structure
within
the
profinite
continuum.
In
§1,
§2,
we
study
two
different
aspects
of
this
topic
of
compari-
son
of
pro-Σ
and
discrete
structures.
Both
§1
and
§2
follow
the
same
pattern:
we
begin
by
proving
an
abstract
and
somewhat
technical
com-
binatorial
result
and
then
proceed
to
discuss
various
applications
of
this
combinatorial
result.
In
§1,
the
main
technical
combinatorial
result
is
summarized
in
The-
orem
A
below
[where
Σ
is
allowed
to
be
an
arbitrary
nonempty
set
of
COMBINATORIAL
ANABELIAN
TOPICS
IV
3
prime
numbers].
This
result
consists
of
versions
of
the
section
con-
jecture
and
Grothendieck
conjecture
—
i.e.,
the
central
issues
of
concern
in
anabelian
geometry
—
for
outer
representations
of
ENN-type
[cf.
Definition
1.7,
(i)].
Here,
we
remark
that
outer
repre-
sentations
of
ENN-type
are
generalizations
of
the
outer
representations
of
NN-type
studied
in
[NodNon].
Just
as
an
outer
representation
of
NN-type
may
be
described,
roughly
speaking,
as
a
purely
combinato-
rial
object
modeled
on
the
outer
Galois
representation
arising
from
a
hyperbolic
curve
over
a
complete
discretely
valued
field
whose
residue
field
is
separably
closed,
an
outer
representation
of
ENN-type
may
be
described,
again
roughly
speaking,
as
an
analogous
sort
of
purely
com-
binatorial
object
that
arises
in
the
case
where
the
residue
field
is
not
necessarily
separably
closed.
The
pro-Σ
section
conjecture
portion
of
Theorem
A
[i.e.,
Theorem
1.13,
(i)]
is
then
obtained
by
combining
•
the
essential
uniqueness
of
fixed
points
of
certain
group
actions
on
profinite
graphs
given
in
[NodNon],
Proposition
3.9,
(i),
(ii),
(iii),
with
•
an
essentially
classical
result
concerning
the
existence
of
fixed
points
[cf.
Lemma
1.6;
Remarks
1.6.1,
1.6.2],
which
amounts,
in
essence,
to
a
geometric
reformulation
of
the
well-known
fact
that
free
pro-Σ
groups
are
torsion-free
[cf.
Remarks
1.13.1;
1.15.2,
(i)].
The
argument
applied
to
prove
this
pro-Σ
section
conjecture
portion
of
Theorem
A
is
essentially
similar
to
the
argument
applied
in
the
tem-
pered
case
discussed
in
[SemiAn],
Theorems
3.7,
5.4,
which
is
reviewed
[in
slightly
greater
generality]
in
the
tempered
section
conjecture
por-
tion
of
Theorem
A
[cf.
Theorem
1.13,
(ii)].
These
section
conjecture
portions
of
Theorem
A
imply,
under
suitable
conditions,
that
there
is
a
natural
bijection
between
conjugacy
classes
of
pro-Σ
and
tempered
sections
[cf.
Theorem
1.13,
(iii)].
This
implication
may
be
regarded
as
an
important
example
of
the
phenomenon
discussed
above,
i.e.,
that
considerations
concerning
sections
are
closely
related
to
the
topic
of
comparison
of
pro-Σ
and
discrete
structures.
Finally,
by
combining
the
pro-Σ
section
conjecture
portion
of
Theorem
A
with
the
combinatorial
version
of
the
Grothendieck
conjecture
obtained
in
[CbTpII],
Theorem
1.9,
(i),
one
obtains
the
Grothendieck
conjecture
portion
of
Theorem
A
[cf.
Corollary
1.14].
Theorem
A
(Combinatorial
versions
of
the
section
conjecture
and
Grothendieck
conjecture).
Let
Σ
be
a
nonempty
set
of
prime
numbers,
G
a
semi-graph
of
anabelioids
of
pro-Σ
PSC-type,
G
a
profi-
nite
group,
and
ρ
:
G
→
Aut(G)
a
continuous
homomorphism
that
is
of
ENN-type
for
a
conducting
subgroup
I
G
⊆
G
[cf.
Definition
1.7,
(i)].
Write
Π
G
for
the
[pro-Σ]
fundamental
group
of
G
and
Π
tp
G
for
the
4
YUICHIRO
HOSHI
AND
SHINICHI
MOCHIZUKI
tempered
fundamental
group
of
G
[cf.
[SemiAn],
Example
2.10;
the
dis-
cussion
preceding
[SemiAn],
Proposition
3.6].
[Thus,
we
have
a
natural
outer
injection
Π
tp
G
→
Π
G
—
cf.
[CbTpIII],
Lemma
3.2,
(i);
the
proof
of
def
out
[CbTpIII],
Proposition
3.3,
(i),
(ii).]
Write
Π
G
=
Π
G
G
[cf.
the
dis-
def
out
tp
cussion
entitled
“Topological
groups”
in
[CbTpI],
§0];
Π
tp
G
=
Π
G
G;
G
→
G,
G
tp
→
G
for
the
universal
pro-Σ
and
pro-tempered
coverings
of
G
corresponding
to
Π
G
,
Π
tp
G
;
VCN(−)
for
the
set
of
vertices,
cusps,
and
nodes
of
the
underlying
[pro-]semi-graph
of
a
[pro-]semi-graph
of
an-
abelioids
[cf.
Definition
1.1,
(i)].
Thus,
we
have
a
natural
commutative
diagram
tp
1
−−−→
Π
tp
G
−−−→
Π
G
−−−→
G
−−−→
1
⏐
⏐
⏐
⏐
1
−−−→
Π
G
−−−→
Π
G
−−−→
G
−−−→
1
—
where
the
horizontal
sequences
are
exact,
and
the
vertical
arrows
tp
are
outer
injections;
Π
tp
G
acts
naturally
on
G
;
Π
G
acts
naturally
on
Then
the
following
hold:
G.
(i)
Suppose
that
ρ
is
l-cyclotomically
full
[cf.
Definition
1.7,
(ii)]
for
some
l
∈
Σ.
Let
s
:
G
→
Π
G
be
a
continuous
section
of
the
natural
surjection
Π
G
G.
Then,
relative
to
the
action
of
via
conjugation
of
VCN-subgroups,
the
image
Π
G
on
VCN(
G)
of
s
stabilizes
some
element
of
VCN(
G).
tp
(ii)
Let
s
:
G
→
Π
G
be
a
continuous
section
of
the
natural
surjec-
tp
tp
tion
Π
tp
G
G.
Then,
relative
to
the
action
of
Π
G
on
VCN(
G
)
via
conjugation
of
VCN-subgroups
[cf.
Definition
1.9],
the
im-
age
of
s
stabilizes
some
element
of
VCN(
G
tp
).
(iii)
Write
Sect(Π
G
/G)
for
the
set
of
Π
G
-conjugacy
classes
of
con-
tinuous
sections
of
the
natural
surjective
homomorphism
Π
G
tp
G
and
Sect(Π
tp
G
/G)
for
the
set
of
Π
G
-conjugacy
classes
of
continuous
sections
of
the
natural
surjective
homomorphism
Π
tp
G
G.
Then
the
natural
map
Sect(Π
tp
G
/G)
−→
Sect(Π
G
/G)
is
injective.
If,
moreover,
ρ
is
l-cyclotomically
full
for
some
l
∈
Σ,
then
this
map
is
bijective.
(iv)
Let
H
be
a
semi-graph
of
anabelioids
of
pro-Σ
PSC-type,
H
a
profinite
group,
ρ
H
:
H
→
Aut(H)
a
continuous
homomor-
phism
that
is
of
ENN-type
for
a
conducting
subgroup
I
H
⊆
H.
Write
Π
H
for
the
[pro-Σ]
fundamental
group
of
H.
Suppose
further
that
ρ
is
verticially
quasi-split
[cf.
Defini-
∼
tion
1.7,
(i)].
Let
β
:
G
→
H
be
a
continuous
isomorphism
def
such
that
β(I
G
)
=
I
H
;
l
∈
Σ
a
prime
number
such
that
ρ
G
=
ρ
COMBINATORIAL
ANABELIAN
TOPICS
IV
5
∼
and
ρ
H
are
l-cyclotomically
full;
α
:
Π
G
→
Π
H
a
continuous
isomorphism
such
that
the
diagram
G
ρ
G
−→
Aut(G)
−→
Out(Π
G
)
|
|
↓
β
↓
H
ρ
H
−→
Aut(H)
−→
Out(Π
H
)
—
where
the
right-hand
vertical
arrow
is
the
isomorphism
ob-
tained
by
conjugating
by
α
—
commutes.
Then
α
is
graphic
[cf.
[CmbGC],
Definition
1.4,
(i)].
The
purely
combinatorial
theory
of
§1
—
i.e.,
the
theory
surrounding
and
including
Theorem
A
—
has
important
applications
to
scheme
theory
—
i.e.,
to
the
theory
of
hyperbolic
curves
over
quite
general
complete
discretely
valued
fields
—
as
follows:
(A-1)
We
observe
that
a
quite
general
result
in
the
style
of
the
main
results
of
[PS]
concerning
valuations
fixed
by
sections
of
the
arithmetic
fundamental
group
follows
formally,
in
the
case
of
hyperbolic
curves
over
quite
general
complete
discretely
valued
fields,
from
Theorem
A
[cf.
Corollary
1.15,
(iii);
Re-
mark
1.15.2,
(i),
(ii)].
The
quite
substantial
generality
of
this
result
is
a
reflection
of
the
purely
combinatorial
nature
of
Theorem
A.
This
approach
contrasts
substantially
with
the
approach
of
[PS]
via
essentially
scheme-theoretic
techniques
such
as
the
local-global
principle
for
the
Brauer
group
[cf.
Re-
mark
1.15.2,
(i)].
The
approach
of
the
present
paper
also
differs
substantially
from
[PS]
in
that
the
transition
from
fixed
points
of
graphs
to
fixed
valuations
is
treated
as
a
formal
consequence
of
well-known
elementary
properties
of
Berkovich
spaces,
i.e.,
in
essence
the
compactness
of
the
unit
interval
[0,
1]
⊆
R
[cf.
Remark
1.15.2,
(ii)].
(A-2)
We
observe
that
the
natural
bijection
between
conjugacy
classes
of
pro-Σ
and
tempered
sections
discussed
in
the
purely
com-
binatorial
setting
of
Theorem
A
implies
a
similar
bijection
in
the
case
of
hyperbolic
curves
over
quite
general
complete
dis-
cretely
valued
fields
[cf.
Corollary
1.15,
(vi)].
This
portion
of
the
theory
was
partially
motivated
by
discussions
between
the
second
author
and
Y.
André.
In
the
context
of
(A-1),
we
remark
that,
in
the
Appendix
to
the
present
paper,
we
give
an
elementary
exposition
from
the
point
of
view
of
two-
dimensional
log
regular
log
schemes
of
the
phenomenon
of
conver-
gence
of
valuations,
without
applying
the
language
or
notions,
such
6
YUICHIRO
HOSHI
AND
SHINICHI
MOCHIZUKI
as
Stone-Čech
compactifications,
typically
applied
in
expositions
of
the
theory
of
Berkovich
spaces.
In
§2,
we
turn
to
the
task
of
formulating
discrete
analogues
of
a
substantial
portion
of
the
theory
developed
in
earlier
papers.
This
formulation
centers
around
the
notion
of
a
semi-graph
of
temper-
oids
of
HSD-type
[i.e.,
“hyperbolic
surface
decomposition
type”
—
cf.
Definition
2.3,
(iii)],
which
may
be
thought
of
as
a
natural
discrete
analogue
of
the
notion
of
a
semi-graph
of
anabelioids
of
pro-Σ
PSC-
type
[cf.
[CmbGC],
Definition
1.1,
(i)].
As
the
name
suggests,
this
notion
may
be
thought
of
as
referring
to
the
sort
of
collection
of
dis-
crete
combinatorial
data
that
one
may
associate
to
a
decomposition
of
a
hyperbolic
surface
into
hyperbolic
subsurfaces.
Alternatively,
it
may
be
thought
of
as
referring
to
the
sort
of
collection
of
combinatorial
data
that
arises
from
systems
of
topological
coverings
of
the
system
of
topological
spaces
naturally
associated
to
a
stable
log
curve
over
a
log
point
whose
underlying
scheme
is
the
spectrum
of
the
field
of
complex
numbers
[cf.
Example
2.4,
(i)].
After
discussing
various
basic
proper-
ties
and
terms
related
to
semi-graphs
of
temperoids
of
HSD-type
[cf.
Proposition
2.5;
Definitions
2.6,
2.7],
we
observe
that
the
fundamen-
tal
operations
of
restriction,
partial
compactification,
resolution,
and
generization
discussed
in
[CbTpI],
§2,
admit
natural
compatible
analogues
for
semi-graphs
of
temperoids
of
HSD-type
[cf.
Definitions
2.8,
2.9;
Proposition
2.10].
The
main
technical
combinatorial
result
of
§2
is
summarized
in
The-
orem
B
below.
This
result
asserts,
in
effect,
that
discrete
subgroups
of
the
discrete
fundamental
group
of
a
semi-graph
of
temperoids
of
HSD-
type
satisfy
various
properties
of
interest
if
and
only
if
the
profinite
completions
of
these
discrete
subgroups
satisfy
analogous
properties
[cf.
Theorem
2.15;
Corollary
2.19,
(i)].
The
main
technical
tool
that
is
applied
in
order
to
derive
this
result
is
the
fact
that
any
inclusion
of
a
finitely
generated
group
into
a
[finitely
generated]
free
discrete
group
is,
after
possibly
passing
to
a
suitable
finite
index
subgroup,
necessarily
split
[cf.
[SemiAn],
Corollary
1.6,
(ii),
which
is
applied
in
the
proof
of
Lemma
2.14,
(i),
of
the
present
paper].
Here,
we
recall
that
in
[SemiAn],
this
fact
[i.e.,
[SemiAn],
Corollary
1.6,
(ii)]
is
obtained
as
an
immedi-
ate
consequence
of
“Zariski’s
main
theorem
for
semi-graphs”
[cf.
[SemiAn],
Theorem
1.2].
Theorem
B
(Profinite
versus
discrete
subgroups).
Let
G,
H
be
semi-graphs
of
temperoids
of
HSD-type
[cf.
Definition
2.3,
(iii)].
H
for
the
semi-graphs
of
anabelioids
of
pro-Primes
PSC-
Write
G,
type
determined
by
G,
H
[cf.
Proposition
2.5,
(iii),
in
the
case
where
Σ
=
Primes],
respectively;
Π
G
,
Π
H
for
the
respective
fundamental
groups
of
G,
H
[cf.
Proposition
2.5,
(i)];
Π
G
,
Π
H
for
the
respective
H.
Then
the
following
hold:
[profinite]
fundamental
groups
of
G,
COMBINATORIAL
ANABELIAN
TOPICS
IV
7
(i)
Let
H,
J
⊆
Π
G
be
subgroups.
Since
Π
G
injects
into
its
pro-
l
completion
for
any
l
∈
Primes
[cf.
Remark
2.5.1],
let
us
regard
subgroups
of
Π
G
as
subgroups
of
the
profinite
completion
G
of
Π
G
.
Write
H,
J
⊆
Π
G
for
the
closures
of
H,
J
in
Π
G
,
Π
respectively.
Suppose
that
the
following
conditions
are
satisfied:
(a)
The
subgroups
H
and
J
are
finitely
generated.
(b)
If
J
is
of
infinite
index
in
Π
G
,
then
J
is
of
infinite
G
.
index
in
Π
[Here,
we
note
that
condition
(b)
is
automatically
satisfied
when-
ever
Cusp(G)
=
∅
—
cf.
[SemiAn],
Corollary
1.6,
(ii).]
Then
the
following
hold:
(1)
It
holds
that
J
=
J
∩
Π
G
.
G
such
that
(2)
Suppose
that
there
exists
an
element
γ
∈
Π
−1
.
H
⊆
γ
·
J
·
γ
Then
there
exists
an
element
δ
∈
Π
G
such
that
H
⊆
δ
·
J
·
δ
−1
.
(ii)
Let
∼
α
:
Π
G
−→
Π
H
∼
be
an
outer
isomorphism.
Write
α
:
Π
G
→
Π
H
for
the
outer
isomorphism
determined
by
α
and
the
natural
outer
isomor-
∼
∼
G
→
H
→
phisms
Π
Π
G
,
Π
Π
H
of
Proposition
2.5,
(iii).
Then
the
outer
isomorphism
α
is
group-theoretically
ver-
ticial
(respectively,
group-theoretically
cuspidal;
group-
theoretically
nodal;
graphic)
[cf.
Definition
2.7,
(i),
(ii)]
if
and
only
if
the
outer
isomorphism
α
is
group-theoretically
verticial
[cf.
[CmbGC],
Definition
1.4,
(iv)]
(respectively,
group-theoretically
cuspidal
[cf.
[CmbGC],
Definition
1.4,
(iv)];
group-theoretically
nodal
[cf.
[NodNon],
Definition
1.12];
graphic
[cf.
[CmbGC],
Definition
1.4,
(i)]).
The
significance
of
Theorem
B
lies
in
the
fact
that
it
renders
possi-
ble
a
fairly
straightforward
translation
of
a
substantial
portion
of
the
profinite
results
obtained
in
earlier
papers
by
the
authors
into
discrete
versions,
as
follows:
(B-1)
the
partial
combinatorial
cuspidalization
obtained
in
[CbTpI],
Theorem
A;
[CbTpII],
Theorem
A
[cf.
Corollary
2.20
of
the
present
paper];
(B-2)
the
theory
of
Dehn
multi-twists
summarized
in
[CbTpI],
Theorem
B
[cf.
Corollary
2.21
of
the
present
paper];
(B-3)
the
theory
of
the
tripod
homomorphism
and
metric-admissibility
summarized
in
[CbTpII],
Theorem
C;
[CbTpIII],
Theorems
A,
C,
D
[cf.
Theorem
2.24
of
the
present
paper];
8
YUICHIRO
HOSHI
AND
SHINICHI
MOCHIZUKI
(B-4)
the
archimedean
analogue
[cf.
Corollary
2.25
of
the
present
paper]
of
the
characterization,
given
in
[CbTpIII],
Theorem
B,
of
nonarchimedean
local
Galois
groups
in
the
global
Galois
image
associated
to
a
hyperbolic
curve.
Finally,
in
§3,
we
examine
the
theory
of
canonical
liftings
of
cy-
cles
discussed
in
[Bgg2]
from
the
point
of
view
of
the
profinite
theory
developed
so
far
by
the
authors.
This
approach
contrasts
substan-
tially
with
the
intuitive
topological
approach
of
[Bgg2]
in
the
discrete
case.
From
a
naive
topological
point
of
view,
the
canonical
liftings
of
cycles
in
question
amount
to
once-punctured
tubular
neighbor-
hoods
of
the
given
cycles
[cf.
Figure
1
below],
i.e.,
to
the
construction
of
a
tripod
[i.e.,
a
copy
of
the
projective
line
minus
three
points]
canon-
ically
and
functorially
associated
to
the
cycle.
This
tripod
satisfies
a
remarkable
rigidity
property,
i.e.,
it
admits
a
canonical
isomor-
phism,
subject
to
almost
no
indeterminacies,
with
a
given
fixed
tripod
that
is
independent
of
the
choice
of
the
cycle.
Moreover,
this
canonical
isomorphism
is
functorial
with
respect
to
“geometric”
outer
automor-
phisms
of
the
profinite
fundamental
group
of
the
stable
log
curve
under
consideration
that
lift
to
automorphisms
of
the
profinite
fundamental
group
of
a
configuration
space
[associated
to
the
stable
log
curve]
of
sufficiently
high
dimension.
Here,
by
“geometric”,
we
mean
that
the
outer
automorphism
under
consideration
lies
in
the
kernel
of
the
tripod
homomorphism
studied
in
[CbTpII],
§3.
Indeed,
this
remarkable
rigid-
ity
property
is
obtained
as
an
immediate
consequence
of
the
theory
of
tripod
synchronization
developed
in
[CbTpII],
§3.
The
profinite
version
of
the
theory
of
canonical
liftings
of
cycles
developed
in
§3
is
summarized
in
Theorem
C
below
[cf.
Theorem
3.10].
By
applying
the
translation
apparatus
developed
in
§2
to
this
profinite
version
of
the
theory,
we
also
obtain
a
corresponding
discrete
version
of
the
theory
of
canonical
liftings
of
cycles
[cf.
Theorem
3.14].
Theorem
C
(Canonical
liftings
of
cycles).
Let
(g,
r)
be
a
pair
of
nonnegative
integers
such
that
2g
−
2
+
r
>
0;
Σ
a
set
of
prime
numbers
which
is
either
equal
to
the
entire
set
of
prime
numbers
or
of
cardinality
one;
k
an
algebraically
closed
field
of
characteristic
∈
Σ;
def
log
def
log
S
=
Spec(k)
the
log
scheme
obtained
by
equipping
S
=
Spec(k)
with
the
log
structure
determined
by
the
fs
chart
N
→
k
that
maps
1
→
0;
X
log
=
X
1
log
a
stable
log
curve
of
type
(g,
r)
over
S
log
.
For
positive
integers
m
≤
n,
write
X
n
log
for
the
n-th
log
configuration
space
of
the
stable
log
curve
X
log
[cf.
the
discussion
entitled
“Curves”
in
[CbTpI],
§0];
Π
n
COMBINATORIAL
ANABELIAN
TOPICS
IV
9
for
the
maximal
pro-Σ
quotient
of
the
kernel
of
the
natural
surjection
π
1
(X
n
log
)
π
1
(S
log
);
log
log
p
log
n/m
:
X
n
−→
X
m
,
p
Π
n/m
:
Π
n
Π
m
,
def
Π
n/m
=
Ker(p
Π
G,
Π
G
n/m
)
⊆
Π
n
,
for
the
objects
defined
in
the
discussion
at
the
beginning
of
[CbTpII],
§3;
[CbTpII],
Definition
3.1.
Let
I
⊆
Π
2/1
⊆
Π
2
be
a
cuspidal
inertia
group
associated
to
the
diagonal
cusp
of
a
fiber
of
p
log
2/1
;
Π
tpd
⊆
Π
3
a
3-central
{1,
2,
3}-tripod
of
Π
3
[cf.
[CbTpII],
Definition
3.7,
(ii)];
I
tpd
⊆
Π
tpd
a
cuspidal
subgroup
of
Π
tpd
that
does
not
arise
from
a
∗
∗∗
cusp
of
a
fiber
of
p
log
3/2
;
J
tpd
,
J
tpd
⊆
Π
tpd
cuspidal
subgroups
of
Π
tpd
∗
∗∗
such
that
I
tpd
,
J
tpd
,
and
J
tpd
determine
three
distinct
Π
tpd
-conjugacy
classes
of
closed
subgroups
of
Π
tpd
.
[Note
that
one
verifies
immediately
from
the
various
definitions
involved
that
such
cuspidal
subgroups
I
tpd
,
∗
∗∗
,
and
J
tpd
always
exist.]
For
positive
integers
n
≥
2,
m
≤
n
and
J
tpd
FC
α
∈
Aut
(Π
n
)
[cf.
[CmbCsp],
Definition
1.1,
(ii)],
write
α
m
∈
Aut
FC
(Π
m
)
for
the
automorphism
of
Π
m
determined
by
α;
Aut
FC
(Π
n
,
I)
⊆
Aut
FC
(Π
n
)
for
the
subgroup
consisting
of
β
∈
Aut
FC
(Π
n
)
such
that
β
2
(I)
=
I;
Aut
FC
(Π
n
)
G
⊆
Aut
FC
(Π
n
)
for
the
subgroup
consisting
of
β
∈
Aut
FC
(Π
n
)
such
that
the
image
of
β
via
the
composite
Aut
FC
(Π
n
)
Out
FC
(Π
n
)
→
Out
FC
(Π
1
)
→
Out(Π
G
)
—
where
the
second
arrow
is
the
natural
injection
of
[NodNon],
Theo-
rem
B,
and
the
third
arrow
is
the
homomorphism
induced
by
the
natural
∼
outer
isomorphism
Π
1
→
Π
G
—
is
graphic
[cf.
[CmbGC],
Definition
1.4,
(i)];
def
Aut
FC
(Π
n
,
I)
G
=
Aut
FC
(Π
n
,
I)
∩
Aut
FC
(Π
n
)
G
;
Cycle
n
(Π
1
)
for
the
set
of
n-cuspidalizable
cycle-subgroups
of
Π
1
[cf.
Defini-
tion
3.5,
(i),
(ii)];
Tpd
I
(Π
2/1
)
for
the
set
of
closed
subgroups
T
⊆
Π
2/1
such
that
T
is
a
tripodal
sub-
group
associated
to
some
2-cuspidalizable
cycle-subgroup
of
Π
1
[cf.
Definition
3.6,
(i)],
and,
moreover,
I
is
a
distinguished
cuspidal
subgroup
[cf.
Definition
3.6,
(ii)]
of
T
.
Then
the
following
hold:
(i)
Let
n
≥
3
be
a
positive
integer.
Then
Aut
FC
(Π
n
,
I)
G
acts
naturally
on
Cycle
n
(Π
1
),
Tpd
I
(Π
2/1
);
there
exists
a
unique
Aut
FC
(Π
n
,
I)
G
-equivariant
map
C
I
:
Cycle
n
(Π
1
)
−→
Tpd
I
(Π
2/1
)
10
YUICHIRO
HOSHI
AND
SHINICHI
MOCHIZUKI
such
that,
for
every
J
∈
Cycle
n
(Π
1
),
C
I
(J)
is
a
tripodal
sub-
group
associated
to
J
[cf.
Definition
3.6,
(i)].
Moreover,
there
exists
an
assignment
Cycle
n
(Π
1
)
J
→
syn
I,J
—
where
syn
I,J
denotes
an
I-conjugacy
class
of
isomorphisms
∼
Π
tpd
→
C
I
(J)
—
such
that
(a)
syn
I,J
maps
I
tpd
bijectively
onto
I,
∗
∗∗
(b)
syn
I,J
maps
the
subgroups
J
tpd
,
J
tpd
bijectively
onto
lift-
ing
cycle-subgroups
of
C
I
(J)
[cf.
Definition
3.6,
(ii)],
and
(c)
for
α
∈
Aut
FC
(Π
n
,
I)
G
,
the
diagram
[of
I
tpd
-,
I-conjugacy
classes
of
isomorphisms]
Π
tpd
−−−→
⏐
⏐
syn
I,J
Π
tpd
⏐
⏐
syn
I,α
(J)
1
C
I
(J)
−−−→
C
I
(α
1
(J))
—
where
the
upper
horizontal
arrow
is
the
[uniquely
de-
termined
—
cf.
the
commensurable
terminality
of
I
tpd
in
Π
tpd
discussed
in
[CmbGC],
Proposition
1.2,
(ii)]
I
tpd
-
conjugacy
class
of
automorphisms
of
Π
tpd
that
lifts
T
Π
tpd
(α)
[cf.
[CbTpII],
Definition
3.19]
and
preserves
I
tpd
;
the
lower
horizontal
arrow
is
the
I-conjugacy
class
of
isomorphisms
induced
by
α
2
[cf.
the
“Aut
FC
(Π
n
,
I)
G
-equivariance”
men-
tioned
above]
—
commutes
up
to
possible
composition
with
the
cycle
symmetry
of
C
I
(α
1
(J))
associated
to
I
[cf.
Definition
3.8].
Finally,
the
assignment
J
→
syn
I,J
is
uniquely
determined,
up
to
possible
composition
with
cy-
cle
symmetries,
by
these
conditions
(a),
(b),
and
(c).
(ii)
Let
n
≥
4
be
a
positive
integer,
α
∈
Aut
FC
(Π
n
,
I)
G
,
and
J
∈
Cycle
n
(Π
1
).
Then
there
exists
an
automorphism
β
∈
Aut
FC
(Π
n
,
I)
G
such
that
the
FC-admissible
outer
automorphism
of
Π
3
determined
by
β
3
lies
in
the
kernel
of
the
tripod
homo-
morphism
T
Π
tpd
of
[CbTpII],
Definition
3.19,
and,
moreover,
α
1
(J)
=
β
1
(J).
Finally,
the
diagram
[of
I
tpd
-,
I-conjugacy
classes
of
isomorphisms]
Π
tpd
⏐
⏐
syn
I,J
Π
tpd
⏐
⏐
syn
I,α
(J)
=syn
I,β
(J)
1
1
C
I
(J)
−−−→
C
I
(α
1
(J))
=
C
I
(β
1
(J))
COMBINATORIAL
ANABELIAN
TOPICS
IV
11
—
where
the
lower
horizontal
arrow
is
the
isomorphism
in-
duced
by
β
2
[cf.
the
“Aut
FC
(Π
n
,
I)
G
-equivariance”
mentioned
in
(i)]
—
commutes
up
to
possible
composition
with
the
cycle
symmetry
of
C
I
(α
1
(J))
=
C
I
(β
1
(J))
associated
to
I.
a
cycle
lifting
cycles
Figure
1:
A
cycle
and
lifting
cycles
Acknowledgment
The
authors
would
like
to
thank
Yu
Iijima
and
Yu
Yang
for
pointing
out
minor
errors
in
an
earlier
version
of
the
present
paper.
The
first
author
was
supported
by
Grant-in-Aid
for
Scientific
Research
(C),
No.
24540016,
Japan
Society
for
the
Promotion
of
Science.
12
YUICHIRO
HOSHI
AND
SHINICHI
MOCHIZUKI
0.
Notations
and
Conventions
Sets:
Let
S
be
a
finite
set.
Then
we
shall
write
S
for
the
cardinality
of
S.
Let
S
be
a
set
equipped
with
an
action
by
a
group
G.
Then
we
shall
write
S
G
⊆
S
for
the
subset
consisting
of
elements
of
S
fixed
by
the
action
of
G
on
S.
Numbers:
Write
Primes
for
the
set
of
all
prime
numbers.
Let
Σ
be
a
set
of
prime
numbers.
Then
we
shall
refer
to
a
nonzero
integer
n
as
a
Σ-integer
if
every
prime
divisor
of
n
is
contained
in
Σ.
The
notation
R
will
be
used
to
denote
the
set,
additive
group,
or
field
of
real
numbers,
each
of
which
we
regard
as
being
equipped
with
its
usual
topology.
The
notation
C
will
be
used
to
denote
the
set,
additive
group,
or
field
of
complex
numbers,
each
of
which
we
regard
as
being
equipped
with
its
usual
topology.
Groups:
Let
Σ
be
a
set
of
prime
numbers
and
f
:
G
→
H
a
homomor-
phism
(respectively,
outer
homomorphism)
of
groups.
Then
we
shall
say
that
f
is
Σ-compatible
if
the
homomorphism
(respectively,
outer
homomorphism)
f
Σ
:
G
Σ
→
H
Σ
between
pro-Σ
completions
induced
by
f
is
injective.
Note
that
one
verifies
easily
that
if
G
is
a
group,
and
H
⊆
G
is
a
subgroup
of
G
of
finite
index,
then
the
natural
inclusion
H
→
G
is
Primes-compatible.
If
G
is
a
topological
group,
then
we
shall
write
G
ab
for
the
abelianization
of
G,
i.e.,
the
quotient
of
G
by
the
closed
normal
subgroup
of
G
generated
by
the
commutators
of
G.
If
G
is
a
profinite
group,
then
we
shall
write
G
G
Σ-ab-free
for
the
maximal
pro-Σ
abelian
torsion-free
quotient
of
G.
We
shall
use
the
terms
normally
terminal
and
commensurably
terminal
as
they
are
defined
in
the
discussion
entitled
“Topological
groups”
in
[CbTpI],
§0.
If
I,
J
⊆
G
are
closed
subgroups
of
a
topological
group
G,
then
we
shall
write
I
≺
J
if
some
open
subgroup
of
I
is
contained
in
J.
COMBINATORIAL
ANABELIAN
TOPICS
IV
13
1.
The
combinatorial
section
conjecture
In
the
present
§1,
we
study
outer
representations
of
ENN-type
[cf.
Definition
1.7,
(i),
below]
on
the
fundamental
group
of
a
semi-graph
of
anabelioids
of
PSC-type
[cf.
[CmbGC],
Definition
1.1,
(i)].
Roughly
speaking,
such
outer
representations
may
be
thought
of
as
an
abstract
combinatorial
version
of
the
natural
outer
representation
of
the
maxi-
mal
tamely
ramified
quotient
of
the
absolute
Galois
group
of
a
complete
local
field
on
the
logarithmic
fundamental
group
of
the
geometric
spe-
cial
fiber
of
a
stable
model
of
a
pointed
stable
curve
over
the
complete
local
field.
By
comparison
to
the
outer
representation
of
NN-type
stud-
ied
in
[NodNon],
outer
representations
of
ENN-type
correspond
to
the
situation
in
which
the
residue
field
of
the
complete
local
field
under
consideration
is
not
necessarily
separably
closed.
Such
outer
represen-
tations
of
ENN-type
give
rise
to
a
surjection
of
profinite
groups,
which
corresponds,
in
the
case
of
pointed
stable
curves
over
complete
local
fields,
to
the
surjection
from
the
arithmetic
fundamental
group
to
[some
quotient
of]
the
absolute
Galois
group
of
the
base
field.
Our
first
main
result
[cf.
Theorem
1.13,
(i),
below]
asserts
that,
under
the
additional
assumption
that
the
associated
cyclotomic
character
has
open
image,
any
section
of
this
surjection
necessarily
admits
a
fixed
point
[i.e.,
a
fixed
vertex
or
edge].
This
“combinatorial
section
conjecture”
is
ob-
tained
as
an
immediate
consequence
of
an
essentially
classical
result
concerning
fixed
points
of
group
actions
on
graphs
[cf.
Lemma
1.6
be-
low].
By
applying
this
existence
of
fixed
points,
we
show
that
there
is
a
natural
bijection
between
conjugacy
classes
of
profinite
sections
and
conjugacy
classes
of
tempered
sections
[cf.
Theorem
1.13,
(iii),
below]
and
derive
a
rather
strong
version
of
the
combinatorial
Grothendieck
conjecture
[cf.
[NodNon],
Theorem
A;
[CbTpII],
Theorem
1.9]
for
cy-
clotomically
full
outer
representations
of
ENN-type
[cf.
Corollary
1.14
below].
We
also
observe
in
passing
that
a
generalization
of
the
main
result
of
[PS]
may
be
obtained
as
a
consequence
of
the
theory
discussed
in
the
present
§1
[cf.
Corollary
1.15
below].
In
the
present
§1,
let
Σ
be
a
nonempty
set
of
prime
numbers
and
G
a
semi-graph
of
anabelioids
of
pro-Σ
PSC-type
[cf.
[CmbGC],
Definition
1.1,
(i)].
Write
G
for
the
underlying
semi-graph
of
G,
Π
G
for
the
[pro-Σ]
fundamental
group
of
G,
and
Π
tp
G
for
the
tempered
fundamental
group
of
G
[cf.
[SemiAn],
Example
2.10;
the
discussion
preceding
[SemiAn],
Proposition
3.6].
Thus,
we
have
a
natural
outer
injection
Π
tp
G
→
Π
G
—
cf.
[CbTpIII],
Lemma
3.2,
(i);
the
proof
of
[CbTpIII],
Proposition
3.3,
(i),
(ii).
Let
us
write
G
−→
G,
G
tp
−→
G
14
YUICHIRO
HOSHI
AND
SHINICHI
MOCHIZUKI
for
the
universal
pro-Σ
and
pro-tempered
coverings
of
G
corresponding
to
Π
G
,
Π
tp
G
and
VCN(
G
tp
)
=
lim
VCN(H
tp
)
←−
tp
—
where
H
(respectively,
H
)
ranges
over
the
subcoverings
of
G
→
G
(respectively,
G
tp
→
G)
corresponding
to
open
subgroups
of
Π
G
(re-
spectively,
Π
tp
G
),
and
VCN(−)
denotes
the
“VCN(−)”
of
the
under-
lying
semi-graph
of
the
semi-graph
of
anabelioids
in
parentheses
[cf.
Definition
1.1,
(i),
below;
[NodNon],
Definition
1.1,
(iii)].
We
begin
by
reviewing
certain
well-known
facts
concerning
semi-
graphs
and
group
actions
on
semi-graphs.
=
lim
VCN(H),
VCN(
G)
←−
def
def
Definition
1.1.
Let
Γ
be
a
semi-graph
[cf.
the
discussion
at
the
be-
ginning
of
[SemiAn],
§1].
(i)
We
shall
write
Vert(Γ)
(respectively,
Cusp(Γ);
Node(Γ))
for
the
set
of
vertices
(respectively,
open
edges,
i.e.,
“cusps”;
closed
def
edges,
i.e.,
“nodes”)
of
Γ.
We
shall
write
Edge(Γ)
=
Cusp(Γ)
def
Node(Γ);
VCN(Γ)
=
Vert(Γ)
Edge(Γ).
(ii)
We
shall
write
V
Γ
:
Edge(Γ)
−→
2
Vert(Γ)
(respectively,
C
Γ
:
Vert(Γ)
−→
2
Cusp(Γ)
;
N
Γ
:
Vert(Γ)
−→
2
Node(Γ)
;
E
Γ
:
Vert(Γ)
−→
2
Edge(Γ)
)
[cf.
(i);
the
discussion
entitled
“Sets”
in
[CbTpI],
§0]
for
the
map
obtained
by
sending
e
∈
Edge(Γ)
(respectively,
v
∈
Vert(Γ);
v
∈
Vert(Γ);
v
∈
Vert(Γ))
to
the
set
of
vertices
(respectively,
open
edges;
closed
edges;
edges)
of
Γ
to
which
e
abuts
(respectively,
which
abut
to
v;
which
abut
to
v;
which
abut
to
v).
For
sim-
plicity,
we
shall
write
V
(resp
C;
N
;
E)
instead
of
V
Γ
(resp
C
Γ
;
N
Γ
;
E
Γ
)
when
there
is
no
danger
of
confusion.
(iii)
Let
n
be
a
nonnegative
integer;
v,
w
∈
Vert(Γ)
[cf.
(i)].
Then
we
shall
write
δ(v,
w)
≤
n
if
the
following
conditions
are
satis-
fied:
•
If
n
=
0,
then
v
=
w.
•
If
n
≥
1,
then
either
δ(v,
w)
≤
n−1
or
there
exist
n
closed
edges
e
1
,
.
.
.
,
e
n
∈
Node(Γ)
of
Γ
[cf.
(i)]
and
n
+
1
vertices
v
0
,
.
.
.
,
v
n
∈
Vert(Γ)
of
Γ
such
that
v
0
=
v,
v
n
=
w,
and,
for
1
≤
i
≤
n,
it
holds
that
V(e
i
)
=
{v
i−1
,
v
i
}
[cf.
(ii)].
Moreover,
we
shall
write
δ(v,
w)
=
n
if
δ(v,
w)
≤
n
but
δ(v,
w)
≤
n
−
1.
If
δ(v,
w)
=
n,
then
we
shall
say
that
the
distance
be-
tween
v
and
w
is
equal
to
n.
COMBINATORIAL
ANABELIAN
TOPICS
IV
15
Definition
1.2.
Let
Γ
be
a
semi-graph.
(i)
Let
G
be
a
group
that
acts
on
Γ.
Then
[by
a
slight
abuse
of
notation,
relative
to
the
notation
defined
in
the
discussion
entitled
“Sets”
in
§0]
we
shall
write
Γ
G
for
the
semi-graph
[i.e.,
the
“G-invariant
portion
of
Γ”]
defined
as
follows:
•
We
take
Vert(Γ
G
)
to
be
Vert(Γ)
G
[cf.
Definition
1.1,
(i);
the
discussion
entitled
“Sets”
in
§0].
•
We
take
Edge(Γ
G
)
to
be
Edge(Γ)
G
[cf.
Definition
1.1,
(i);
the
discussion
entitled
“Sets”
in
§0].
•
Let
e
∈
Edge(Γ
G
)
=
Edge(Γ)
G
.
Then
the
coincidence
map
ζ
e
:
e
−→
Vert(Γ
G
)
∪
{Vert(Γ
G
)}
of
Γ
G
[cf.
item
(3)
of
the
discussion
at
the
beginning
of
[SemiAn],
§1]
is
defined
as
follows:
Write
ζ
e
Γ
:
e
→
Vert(Γ)
∪
{Vert(Γ)}
for
the
coincidence
map
associated
to
Γ.
Then,
for
b
∈
e,
if
b
∈
e
G
and
ζ
e
Γ
(b)
∈
Vert(Γ)
G
(respectively,
if
either
b
∈
e
G
or
ζ
e
Γ
(b)
∈
Vert(Γ)
G
),
then
def
def
we
set
ζ
e
(b)
=
ζ
e
Γ
(b)
(respectively,
=
Vert(Γ
G
)).
In
par-
ticular,
it
holds
that
V
Γ
G
(e)
=
V
Γ
(e)∩Vert(Γ)
G
[cf.
Defini-
tion
1.1,
(ii)]
whenever
it
holds
either
that
Γ
is
untangled
[i.e.,
every
node
abuts
to
two
distinct
vertices
—
cf.
the
discussion
entitled
“Semi-graphs”
in
[NodNon],
§0]
or
that
G
acts
on
Γ
without
inversion
[i.e.,
that
if
e
∈
Edge(Γ)
G
,
then
e
=
e
G
].
(ii)
We
shall
write
Γ
÷
for
the
semi-graph
[i.e.,
the
result
of
“subdividing”
Γ]
defined
as
follows:
•
We
take
Vert(Γ
÷
)
to
be
Vert(Γ)
Edge(Γ).
•
We
take
Edge(Γ
÷
)
to
be
the
set
of
branches
of
Γ.
•
Let
b
be
a
branch
of
an
edge
e
of
Γ.
Write
e(b)
∈
Edge(Γ
÷
),
v(e)
∈
Vert(Γ
÷
)
for
the
edge
and
vertex
of
Γ
÷
correspond-
ing
to
b,
e,
respectively.
If
b
abuts,
relative
to
Γ,
to
a
ver-
tex
v
∈
Vert(Γ),
then
we
take
the
edge
e(b)
to
be
a
node
that
abuts
to
v(e)
and
the
vertex
of
Γ
÷
corresponding
to
v
∈
Vert(Γ).
If
b
does
not
abut,
relative
to
Γ,
to
a
vertex
of
Γ,
then
we
take
the
edge
e(b)
to
be
a
cusp
that
abuts
to
v(e).
16
YUICHIRO
HOSHI
AND
SHINICHI
MOCHIZUKI
Definition
1.3.
Let
Γ
be
a
semi-graph
and
Γ
0
⊆
Γ
a
sub-semi-graph
[cf.
[SemiAn],
the
discussion
following
the
figure
entitled
“A
Typical
Semi-graph”]
of
Γ.
(i)
We
shall
write
Γ
0
⊆
Γ
for
the
sub-semi-graph
of
Γ
[i.e.,
whenever
a
suitable
condi-
tion
is
satisfied
[cf.
Lemma
1.4,
(v),
below],
a
sort
of
“open
neighborhood”
of
Γ
0
]
whose
sets
of
vertices
and
edges
are
de-
fined
as
follows.
[Here,
we
recall
that
it
follows
immediately
from
the
definition
of
a
sub-semi-graph
that
a
sub-semi-graph
is
completely
determined
by
its
sets
of
vertices
and
edges.]
•
We
take
Vert(Γ
0
)
to
be
Vert(Γ
0
).
•
We
take
Edge(Γ
0
)
to
be
the
set
of
edges
e
of
Γ
such
that
V
Γ
(e)
∩
Vert(Γ
0
)
=
∅.
(ii)
We
shall
write
Γ
∈
0
⊆
Γ
for
the
sub-semi-graph
of
Γ
whose
sets
of
vertices
and
edges
are
taken
to
be
Vert(Γ)\Vert(Γ
0
),
Edge(Γ)\Edge(Γ
0
),
respectively.
def
(iii)
We
shall
write
Γ
∈
0
=
(Γ
∈
0
)
[cf.
(i),
(ii)].
(iv)
We
shall
say
that
an
edge
e
of
Γ
is
a
Γ
0
-bridge
if
V
Γ
(e)
∩
Vert(Γ
0
),
V
Γ
(e)
∩
Vert(Γ
∈
0
)
=
∅.
[Thus,
one
verifies
easily
that
every
Γ
0
-bridge
is
a
node.]
We
shall
write
Brdg(Γ
0
⊆
Γ)
⊆
Node(Γ)
for
the
set
of
Γ
0
-bridges
of
Γ.
By
abuse
of
notation,
we
shall
write
Brdg(Γ
0
⊆
Γ)
⊆
Γ
for
the
sub-semi-graph
of
Γ
whose
sets
of
vertices
and
edges
are
taken
to
be
∅
[i.e.,
the
empty
set],
Brdg(Γ
0
⊆
Γ)
⊆
Node(Γ),
respectively.
Lemma
1.4
(Basic
properties
of
sub-semi-graphs).
Let
Γ
be
a
semi-graph,
Γ
0
⊆
Γ
a
sub-semi-graph
[cf.
[SemiAn],
the
discussion
fol-
lowing
the
figure
entitled
“A
Typical
Semi-graph”]
of
Γ,
G
a
group,
and
ρ
:
G
→
Aut(Γ)
an
action
of
G
on
Γ.
Then
the
following
hold:
(i)
Suppose
either
that
Γ
is
untangled
or
that
G
acts
on
Γ
with-
out
inversion.
Then
the
semi-graph
Γ
G
[cf.
Definition
1.2,
(i)]
may
be
regarded,
in
a
natural
way,
as
a
sub-semi-graph
of
Γ.
(ii)
Suppose
that
G
acts
on
Γ
without
inversion,
and
that
every
edge
of
Γ
abuts
to
at
least
one
vertex
of
Γ.
Then
every
edge
of
Γ
G
abuts
to
at
least
one
vertex
of
Γ
G
.
(iii)
The
semi-graph
Γ
÷
[cf.
Definition
1.2,
(ii)]
is
untangled.
(iv)
There
exists
a
natural
injection
Aut(Γ)
→
Aut(Γ
÷
).
More-
over,
the
resulting
action
ρ
÷
of
G
on
Γ
÷
[i.e.,
the
composite
ρ
G
→
Aut(Γ)
→
Aut(Γ
÷
)]
is
an
action
without
inversion.
Finally,
it
holds
that
Γ
G
=
∅
if
and
only
if
(Γ
÷
)
G
=
∅.
COMBINATORIAL
ANABELIAN
TOPICS
IV
17
(v)
Suppose
that
every
edge
of
Γ
0
abuts
to
at
least
one
vertex
of
Γ
0
.
Then
Γ
0
may
be
regarded,
in
a
natural
way,
as
a
sub-
semi-graph
of
Γ
0
[cf.
Definition
1.3,
(i)].
(vi)
We
have
an
equality
of
subsets
of
Edge(Γ):
∈
Edge(Γ
0
)
∩
Edge(Γ
0
)
=
Brdg(Γ
0
⊆
Γ).
Proof.
The
assertions
of
Lemma
1.4
follow
immediately
from
the
vari-
ous
definitions
involved.
Lemma
1.5
(Sub-semi-graphs
of
invariants).
In
the
situation
of
Lemma
1.4,
suppose
either
that
Γ
is
untangled
or
that
G
acts
on
Γ
without
inversion.
Suppose,
moreover,
that
the
sub-semi-graph
Γ
0
⊆
Γ
is
a
connected
component
of
the
sub-semi-graph
Γ
G
⊆
Γ
[cf.
Lemma
1.4,
(i)].
Then
the
following
hold:
(i)
The
action
ρ
naturally
determines
actions
of
G
on
Γ
0
,
Γ
0
,
∈
Γ
0
,
respectively.
(ii)
The
intersection
of
Γ
0
⊆
Γ
with
any
connected
component
of
Γ
G
⊆
Γ
that
is
=
Γ
0
is
empty.
(iii)
We
have
an
equality
of
subsets
of
Edge(Γ):
Edge(Γ
G
)
∩
Brdg(Γ
0
⊆
Γ)
=
∅.
Proof.
The
assertions
of
Lemma
1.5
follow
immediately
from
the
vari-
ous
definitions
involved.
Lemma
1.6
(Existence
of
fixed
points).
Let
Γ
be
a
finite
con-
nected
[hence
nonempty]
semi-graph,
G
a
finite
solvable
group
whose
order
is
a
Σ-integer
[cf.
the
discussion
entitled
“Numbers”
in
§0],
and
ρ
:
G
−→
Aut(Γ)
for
the
[discrete]
topological
funda-
an
action
of
G
on
Γ.
Write
Π
disc
Γ
Σ
disc
→
Γ,
mental
group
of
Γ;
Π
Γ
for
the
pro-Σ
completion
of
Π
disc
Γ
;
Γ
Σ
→
Γ
for
the
discrete,
pro-Σ
universal
coverings
of
Γ
corresponding
Γ
Σ
to
Π
disc
Γ
,
Π
Γ
,
respectively.
Let
∈
{disc,
Σ}.
Write
Aut(
Γ
→
Γ)
⊆
such
that
α
)
for
the
group
of
automorphisms
α
of
Γ
lies
over
Aut(
Γ
a(n)
[necessarily
unique]
automorphism
α
of
Γ;
→
Γ)
−→
Aut(Γ)
Aut(
Γ
α
→
α
for
the
resulting
natural
homomorphism;
Π
Γ//G
=
Aut(
Γ
→
Γ)
×
Aut(Γ)
G
def
18
YUICHIRO
HOSHI
AND
SHINICHI
MOCHIZUKI
→
Γ)
→
for
the
fiber
product
of
the
natural
homomorphism
Aut(
Γ
Aut(Γ)
and
the
action
ρ
:
G
→
Aut(Γ).
Thus,
one
verifies
easily
that
Π
Γ//G
fits
into
an
exact
sequence
1
−→
Π
Γ
−→
Π
Γ//G
−→
G
−→
1.
Let
s
:
G
→
Π
Γ//G
be
a
section
of
the
above
exact
sequence.
Write
)
for
the
action
obtained
by
forming
the
compos-
ρ
s
:
G
→
Aut(
Γ
pr
1
s
→
Γ)
→
Aut(
Γ
).
We
shall
say
that
a
ite
G
→
Π
→
Aut(
Γ
Γ//G
Σ
→
Γ
is
G-compatible
if
connected
finite
subcovering
Γ
∗
→
Γ
of
Γ
Γ
∗
→
Γ
is
Galois,
and,
moreover,
the
corresponding
normal
open
sub-
Σ
group
of
Π
Σ
Γ
is
preserved
by
the
outer
action
of
G,
via
ρ,
on
Π
Γ
.
If
Σ
→
Γ,
then
Γ
∗
→
Γ
is
a
G-compatible
connected
finite
subcovering
of
Γ
let
us
write
ρ
s,∗
:
G
→
Aut(Γ
∗
)
for
the
action
of
G
on
Γ
∗
determined
by
G
ρ
s
;
Γ
∗
for
the
semi-graph
defined
in
Definition
1.2,
(i),
with
respect
to
the
action
ρ
s,∗
.
[Thus,
if
Γ,
hence
also
Γ
∗
,
is
untangled,
then
Γ
G
∗
is
a
sub-semi-graph
of
Γ
∗
—
cf.
Lemma
1.4,
(i).]
Then
the
following
hold:
(i)
Suppose
that
Γ
is
untangled.
Then,
for
each
G-compatible
Σ
→
Γ,
the
sub-
connected
finite
subcovering
Γ
∗
→
Γ
of
Γ
G
semi-graph
Γ
∗
⊆
Γ
∗
coincides
with
the
disjoint
union
of
some
def
[possibly
empty]
collection
of
connected
components
of
Γ
∗
|
Γ
G
=
Γ
∗
×
Γ
Γ
G
⊆
Γ
∗
.
(ii)
Suppose
that
Γ
is
untangled,
and
that
G
is
isomorphic
to
Z/lZ
for
some
prime
number
l
∈
Σ.
Then,
for
every
G-
Σ
→
Γ,
compatible
connected
finite
subcovering
Γ
∗
→
Γ
of
Γ
the
sub-semi-graph
Γ
G
∗
⊆
Γ
∗
is
nonempty.
disc
)
G
for
the
sub-semi-graph
(iii)
Suppose
that
=
disc.
Write
(
Γ
[cf.
Lemma
1.4,
(i)]
of
[the
necessarily
untangled
semi-graph!]
disc
defined
in
Definition
1.2,
(i),
with
respect
to
the
action
Γ
disc
)
G
is
nonempty
and
connected.
If,
more-
ρ
disc
s
.
Then
(
Γ
over,
we
write
(Γ
G
)
0
⊆
Γ
G
for
the
image
of
the
composite
disc
→
Γ,
then
the
resulting
morphism
(
Γ
disc
)
G
→
disc
)
G
→
Γ
(
Γ
(Γ
G
)
0
is
a
[discrete]
universal
covering
of
(Γ
G
)
0
.
(iv)
Suppose
that
=
disc
(respectively,
=
Σ).
Then
the
set
disc
)
G
VCN(
Γ
Σ
)
G
=
lim
VCN(Γ
∗
)
G
)
(respectively,
VCN(
Γ
←−
def
—
where,
in
the
resp’d
case,
the
projective
limit
is
taken
over
Σ
→
the
G-compatible
connected
finite
subcoverings
Γ
∗
→
Γ
of
Γ
Γ
—
is
nonempty.
(v)
Suppose
that
=
Σ,
that
Γ
is
untangled,
and
that
G
is
isomorphic
to
Z/lZ
for
some
prime
number
l
∈
Σ.
Let
(Γ
G
)
0
⊆
COMBINATORIAL
ANABELIAN
TOPICS
IV
19
Γ
G
be
a
[nonempty]
connected
component
of
Γ
G
such
that
Σ
)
G
→
VCN(Γ)
=
∅
VCN((Γ
G
)
0
)
∩
Im
VCN(
Γ
[cf.
(iv)].
Then
there
exists
a
G-compatible
connected
finite
Σ
→
Γ
such
that
the
image
of
Γ
G
subcovering
Γ
∗
→
Γ
of
Γ
∗
⊆
Γ
∗
G
G
in
Γ
coincides
with
(Γ
)
0
⊆
Γ
.
(vi)
Suppose
that
=
Σ,
and
that
Γ
is
untangled.
Then
the
Σ
determined
by
the
projective
Σ
)
G
of
Γ
sub-pro-semi-graph
(
Γ
system
of
sub-semi-graphs
Γ
G
∗
—
where
Γ
∗
→
Γ
ranges
over
Σ
→
Γ
—
is
the
G-compatible
connected
finite
subcoverings
of
Γ
nonempty
and
connected.
If,
moreover,
we
write
(Γ
G
)
0
⊆
Σ
)
G
→
Γ
Σ
→
Γ,
then
Γ
G
for
the
image
of
the
composite
(
Γ
Σ
)
G
→
(Γ
G
)
0
is
a
pro-Σ
universal
the
resulting
morphism
(
Γ
G
covering
of
(Γ
)
0
.
Proof.
First,
we
verify
assertion
(i).
Let
us
first
observe
that
one
verifies
immediately
that
there
is
an
inclusion
of
sub-semi-graphs
Γ
G
∗
⊆
Γ
∗
|
Γ
G
[cf.
Lemma
1.4,
(i)].
Next,
let
us
observe
that
it
follows
immediately
from
Lemma
1.4,
(iii),
(iv),
that,
by
replacing
Γ
by
Γ
÷
,
we
may
assume
without
loss
of
generality
that
G
acts
without
inversion
on
Γ
[which
implies
that
G
acts
trivially
on
Γ
G
—
cf.
Definition
1.2,
(i)].
Thus,
to
complete
the
verification
of
assertion
(i),
it
suffices
to
verify
that
the
following
assertion
holds:
Claim
1.6.A:
Let
(Γ
∗
|
Γ
G
)
0
⊆
Γ
∗
|
Γ
G
be
a
connected
com-
ponent
of
Γ
∗
|
Γ
G
such
that
(Γ
∗
|
Γ
G
)
0
∩
Γ
∗
G
=
∅.
Then
(Γ
∗
|
Γ
G
)
0
⊆
Γ
G
∗
.
To
verify
Claim
1.6.A,
let
us
observe
that
since
(Γ
∗
|
Γ
G
)
0
∩
Γ
G
∗
=
∅,
the
action
ρ
s,∗
of
G
on
Γ
∗
stabilizes
(Γ
∗
|
Γ
G
)
0
⊆
Γ
∗
.
In
particular,
we
obtain
an
action
of
G
on
(Γ
∗
|
Γ
G
)
0
over
Γ
G
.
Thus,
since
the
action
of
G
on
Γ
G
is
trivial,
and
the
composite
(Γ
∗
|
Γ
G
)
0
→
Γ
∗
|
Γ
G
→
Γ
G
is
a
connected
finite
covering
of
some
connected
component
of
Γ
G
,
again
by
our
assumption
that
(Γ
∗
|
Γ
G
)
0
∩
Γ
G
∗
=
∅,
we
conclude
that
the
action
of
G
on
(Γ
∗
|
Γ
G
)
0
is
trivial,
i.e.,
that
there
is
an
inclusion
of
sub-semi-graphs
(Γ
∗
|
Γ
G
)
0
⊆
Γ
G
∗
.
This
completes
the
proof
of
Claim
1.6.A,
hence
also
of
assertion
(i).
Next,
we
verify
assertion
(ii).
One
verifies
immediately
that
we
may
assume
without
loss
of
generality
that
Γ
∗
=
Γ.
Now
suppose
that
Γ
G
=
∅.
Then
since
G
∼
=
Z/lZ,
it
follows
that
the
action
of
G
on
Γ
is
free,
which
thus
implies
that
the
quotient
map
Γ
Γ/G
is
a
covering
of
Γ/G.
In
particular,
Π
Σ
Γ//G
is
isomorphic
to
the
pro-Σ
completion
of
the
topological
fundamental
group
of
the
semi-graph
Γ/G.
Thus,
the
pro-Σ
group
Π
Σ
Γ//G
is
free,
hence,
in
particular,
torsion-free.
But
this
contradicts
the
existence
of
the
section
of
the
surjection
Π
Σ
Γ//G
G
determined
by
s.
This
completes
the
proof
of
assertion
(ii).
20
YUICHIRO
HOSHI
AND
SHINICHI
MOCHIZUKI
Next,
we
verify
the
resp’d
portion
of
assertion
(iv)
[i.e.,
the
assertion
Σ
)
G
=
∅]
in
the
case
where
G
is
isomorphic
to
Z/lZ
for
some
that
VCN(
Γ
prime
number
l
∈
Σ.
Let
us
first
observe
that
it
follows
immediately
from
Lemma
1.4,
(iii),
(iv),
that,
by
replacing
Γ
by
Γ
÷
,
we
may
assume
without
loss
of
generality
that
Γ
is
untangled.
Thus,
the
assertion
that
Σ
)
G
=
∅
follows
immediately
from
assertion
(ii),
together
with
VCN(
Γ
the
well-known
elementary
fact
that
a
projective
limit
of
nonempty
finite
sets
is
nonempty.
This
completes
the
proof
of
the
assertion
that
Σ
)
G
=
∅
in
the
case
where
G
is
isomorphic
to
Z/lZ
for
some
VCN(
Γ
prime
number
l
∈
Σ.
disc
Next,
we
verify
assertion
(iii).
Let
us
first
observe
that
since
Γ
disc
)
G
is
a
tree,
hence
untangled,
it
follows
from
Lemma
1.4,
(i),
that
(
Γ
disc
.
Next,
let
us
observe
that
it
follows
im-
is
a
sub-semi-graph
of
Γ
mediately
from
Lemma
1.4,
(iv),
that,
by
replacing
Γ
by
Γ
÷
,
we
may
assume
without
loss
of
generality
that
G
acts
without
inversion
on
Γ.
disc
)
G
is
nonempty
and
connected
follows
im-
Thus,
the
assertion
that
(
Γ
mediately
from
[SemiAn],
Lemma
1.8,
(ii).
The
remainder
of
assertion
(iii)
follows
from
a
similar
argument
to
the
argument
applied
in
the
proof
of
assertion
(i).
This
completes
the
proof
of
assertion
(iii).
In
particular,
the
unresp’d
portion
of
assertion
(iv)
[i.e.,
the
assertion
that
disc
)
G
=
∅]
holds.
VCN(
Γ
Next,
we
verify
assertion
(v).
Let
us
first
observe
that,
to
verify
assertion
(v),
it
follows
immediately
from
Lemma
1.4,
(iii),
(iv),
that,
by
replacing
Γ
by
Γ
÷
,
we
may
assume
without
loss
of
generality
that
the
action
ρ
is
an
action
without
inversion,
and
that
every
edge
of
Γ
abuts
to
at
least
one
vertex
of
Γ.
In
particular,
since
[we
have
assumed
that]
(Γ
G
)
0
=
∅,
it
follows
from
Lemma
1.4,
(ii),
(v),
that
(Γ
G
)
0
=
∅
[cf.
Definition
1.3,
(i)].
Now
if
Γ
G
is
connected,
then
one
verifies
imme-
id
diately
that
the
trivial
covering
Γ
→
Γ
satisfies
the
condition
imposed
on
“Γ
∗
→
Γ”
in
the
statement
of
assertion
(v).
Thus,
to
complete
the
verification
of
assertion
(v),
we
may
assume
without
loss
of
generality
that
Γ
G
is
not
connected,
hence
[cf.
Lemma
1.4,
(ii)]
contains
at
least
one
vertex
∈
Vert((Γ
G
)
0
).
In
particular,
(Γ
G
)
∈
0
=
∅
[cf.
Definition
1.3,
(iii)].
Write
((Γ
G
)
)
→
(Γ
G
)
0
0
for
the
trivial
Z/lZ-covering
obtained
by
taking
a
disjoint
union
of
copies
of
(Γ
G
)
0
indexed
by
the
elements
of
G
∈
G
∈
Z/lZ;
((Γ
)
0
)
→
(Γ
)
0
for
the
trivial
Z/lZ-covering
obtained
by
taking
a
disjoint
union
of
copies
of
(Γ
G
)
∈
0
indexed
by
the
elements
G
∈
[cf.
of
Z/lZ.
Then
the
natural
actions
of
G
on
((Γ
G
)
0
)
,
((Γ
)
0
)
G
Lemma
1.5,
(i)]
determine
natural
actions
of
G
×
Z/lZ
on
((Γ
)
)
,
0
G
∈
((Γ
)
0
)
,
i.e.,
we
have
homomorphisms
,
)
ρ
:
G
×
Z/lZ
−→
Aut
((Γ
G
)
0
COMBINATORIAL
ANABELIAN
TOPICS
IV
21
ρ
∈
:
G
×
Z/lZ
−→
Aut
((Γ
G
)
∈
0
)
.
∼
Let
φ
:
G
→
Z/lZ
be
an
isomorphism.
Write
ρ
∈
φ
:
ρ
∈
G
×
Z/lZ
−→
G
×
Z/lZ
−→
Aut
((Γ
G
)
∈
0
)
(a,
b)
→
(a,
φ(a)
+
b)
for
the
composite
of
ρ
∈
with
the
homomorphism
described
in
the
second
line
of
the
display.
def
Next,
for
e
∈
Brdg
=
Brdg((Γ
G
)
0
⊆
Γ)
[cf.
Definition
1.3,
(iv)],
write
G
·
e
⊆
Edge((Γ
G
)
0
)
for
the
G-orbit
of
e.
Then
it
is
immediate
that
G
·
e
⊆
Brdg;
moreover,
since
G
∼
=
Z/lZ,
it
follows
immediately
from
Lemma
1.5,
(iii),
that
G
·
e
is
a
G-torsor.
Next,
let
us
write
def
def
G
((Γ
G
)
G
·
e,
0
)
|
G·e
=
((Γ
)
0
)
×
(Γ
G
)
0
((Γ
G
)
∈
0
)
|
G·e
=
((Γ
G
)
∈
0
)
×
(Γ
G
)
∈
G
·
e
0
[cf.
Lemma
1.4,
(vi)].
Then
one
verifies
easily
from
the
various
defini-
tions
involved
that
the
following
hold:
)
,
((Γ
G
)
∈
0
)
(a)
The
actions
ρ
,
ρ
∈
φ
of
G
×
Z/lZ
on
((Γ
G
)
0
G
∈
determine
actions
on
these
fibers
((Γ
G
)
)
|
,
((Γ
)
)
|
G·e
.
G·e
0
0
G
G
∈
(b)
These
fibers
((Γ
)
0
)
|
G·e
,
((Γ
)
0
)
|
G·e
are
(G×Z/lZ)-torsors
with
respect
to
the
actions
of
(a).
∼
(c)
There
is
a
natural
isomorphism
of
semi-graphs
((Γ
G
)
)
|
G·e
→
0
((Γ
G
)
∈
0
)
|
G·e
[cf.
Lemma
1.4,
(vi)],
which
we
shall
use
to
iden-
tify
these
two
semi-graphs.
G
∈
(d)
Let
e
base
∈
((Γ
G
)
)
|
=
((Γ
)
)
|
G·e
[cf.
(c)]
be
a
lifting
G·e
0
0
of
e
∈
Brdg.
Then
there
is
a
uniquely
determined
[cf.
(b)]
isomorphism
∼
G
∈
ι
e
base
:
((Γ
G
)
0
)
|
G·e
−→
((Γ
)
0
)
|
G·e
of
(G
×
Z/lZ)-torsors
[cf.
(b)]
that
maps
e
base
to
e
base
.
Let
B
be
a
collection
of
elements
“e
base
”
as
in
(d)
such
that
the
map
e
base
→
e
determines
a
bijection
between
B
and
the
set
of
G-orbits
G
∈
of
Brdg.
Thus,
by
gluing
((Γ
G
)
)
to
((Γ
)
)
by
means
of
the
0
0
collection
of
isomorphisms
{ι
e
base
}
e
base
∈B
of
(d)
[cf.
Lemma
1.4,
(vi)],
we
obtain
a
finite
covering
Γ
∗
→
Γ,
together
with
an
action
of
G×Z/lZ
on
Γ
∗
[i.e.,
obtained
by
gluing
the
actions
ρ
,
ρ
∈
φ
],
such
that
the
morphism
Γ
∗
→
Γ
is
equivariant
with
respect
to
this
action
of
G×Z/lZ
on
Γ
∗
and
the
action
of
G
×
Z/lZ
on
Γ
obtained
by
composing
the
projection
G
×
Z/lZ
→
G
with
the
given
action
of
G
on
Γ.
Next,
let
us
observe
that
since
φ
is
an
isomorphism,
and
both
(Γ
G
)
0
and
(Γ
G
)
∈
0
contain
vertices
fixed
by
G
[cf.
the
discussion
at
the
beginning
of
the
present
proof
of
assertion
(v)],
one
verifies
immediately
—
e.g.,
by
considering
the
orbit
by
the
action
of
G
×
{1}
(⊆
G
×
Z/lZ)
of
some
22
YUICHIRO
HOSHI
AND
SHINICHI
MOCHIZUKI
lifting
to
Γ
∗
[which
may
be
chosen
to
pass
through
an
element
of
B]
of
a
path
of
minimal
length
between
such
vertices
fixed
by
G
—
that
Γ
∗
is
connected.
Moreover,
it
follows
from
the
definition
of
Γ
∗
that
the
covering
Γ
∗
→
Γ
is
Galois,
G-compatible,
and
equipped
with
a
natural
∼
Σ
→
Γ
factors
as
a
isomorphism
Gal(Γ
∗
/Γ)
→
Z/lZ;
in
particular,
Γ
Σ
→
Γ
∗
→
Γ.
composite
Γ
Next,
let
us
observe
that,
for
each
g
∈
G,
the
automorphism
α
g
of
Γ
∗
obtained
by
considering
the
difference
between
ρ
s,∗
(g)
and
the
action
of
g
[i.e.,
(g,
0)
∈
G
×
Z/lZ]
on
Γ
∗
defined
above
is
an
automorphism
over
Γ.
Moreover,
it
follows
immediately
from
our
assumption
that
Σ
)
G
→
VCN(Γ)
=
∅
VCN((Γ
G
)
0
)
∩
Im
VCN(
Γ
that
α
g
fixes
an
element
of
VCN(Γ
∗
)
that
maps
to
VCN((Γ
G
)
0
)
⊆
VCN(Γ).
But
this
implies
that
α
g
is
trivial,
i.e.,
that
the
action
ρ
s,∗
of
G
coincides
with
the
action
of
G
(=
G
×
{0}
⊆
G
×
Z/lZ)
on
Γ
∗
defined
above.
On
the
other
hand,
since
φ
is
an
isomorphism,
it
follows
that
(Γ
∗
)
G
⊆
Γ
∗
is
contained
in
the
sub-semi-graph
of
Γ
∗
determined
by
((Γ
G
)
0
)
.
In
particular,
it
follows
immediately
from
Lemma
1.5,
(ii),
that
the
G
G
image
of
Γ
G
∗
⊆
Γ
∗
in
Γ
is
contained
in
(Γ
)
0
⊆
Γ
.
Thus,
it
follows
immediately
from
assertion
(i)
that
the
image
of
Γ
G
∗
⊆
Γ
∗
in
Γ
coincides
G
G
with
(Γ
)
0
⊆
Γ
.
This
completes
the
proof
of
assertion
(v).
Next,
we
verify
assertion
(vi).
First,
we
claim
that
the
following
assertion
holds:
Claim
1.6.B:
If
G
is
isomorphic
to
Z/lZ
for
some
prime
number
l
∈
Σ,
then
assertion
(vi)
holds.
Indeed,
it
follows
from
the
resp’d
portion
of
assertion
(iv)
[i.e.,
the
Σ
)
G
=
∅]
in
the
case
where
G
is
isomorphic
to
assertion
that
VCN(
Γ
Z/lZ
for
some
prime
number
l
∈
Σ
[i.e.,
the
case
that
has
already
Σ
)
G
=
∅.
On
the
other
hand,
it
follows
im-
been
verified!]
that
(
Γ
mediately
from
assertion
(v)
[i.e.,
by
allowing
“Γ”
to
vary
among
the
Σ
→
Γ]
that
(
Γ
Σ
)
G
is
G-compatible
connected
finite
subcoverings
of
Γ
connected.
Thus,
the
final
portion
of
assertion
(vi)
[in
the
case
where
G
is
isomorphic
to
Z/lZ
for
some
prime
number
l
∈
Σ]
follows
imme-
diately
from
assertion
(i)
[and
the
evident
pro-Σ
version
of
[SemiAn],
Proposition
2.5,
(i)].
This
completes
the
proof
of
Claim
1.6.B.
Next,
we
verify
assertion
(vi)
for
arbitrary
finite
solvable
G
by
in-
duction
on
G
.
Let
us
first
observe
that
it
follows
immediately
from
Lemma
1.4,
(iii),
(iv),
that,
by
replacing
Γ
by
Γ
÷
,
we
may
assume
with-
out
loss
of
generality
that
the
action
ρ
is
an
action
without
inversion.
Next,
observe
that
since
G
is
finite
and
solvable,
there
exists
a
normal
subgroup
N
⊆
G
of
G
such
that
G/N
is
a
nontrivial
finite
group
of
prime
order.
Then
it
follows
from
the
induction
hypothesis
that
if
we
COMBINATORIAL
ANABELIAN
TOPICS
IV
23
write
(Γ
N
)
0
⊆
Γ
N
for
the
[nonempty,
connected!]
image
of
the
com-
Σ
→
Γ,
then
the
resulting
morphism
(
Γ
Σ
)
N
→
(Γ
N
)
0
Σ
)
N
→
Γ
posite
(
Γ
is
a
pro-Σ
universal
covering
of
(Γ
N
)
0
,
and,
moreover,
[since
the
action
Σ
)
N
.
Next,
let
ρ
is
an
action
without
inversion]
N
acts
trivially
on
(
Γ
us
observe
that
since
N
is
normal
in
G,
[one
verifies
immediately
that]
Σ
Σ
N
⊆
Γ
Σ
.
Thus,
by
replacing
the
action
ρ
Σ
s
of
G
on
Γ
preserves
(
Γ
)
Σ
)
N
→
(Γ
N
)
0
,
G/N
)
and
applying
Claim
1.6.B,
we
Σ
→
Γ,
G)
by
((
Γ
(
Γ
conclude
that
assertion
(vi)
holds
for
the
given
G.
This
completes
the
proof
of
assertion
(vi).
Finally,
we
verify
the
resp’d
portion
of
assertion
(iv)
[i.e.,
the
as-
Σ
)
G
=
∅].
Let
us
first
observe
that,
to
verify
the
sertion
that
VCN(
Γ
Σ
)
G
=
∅,
it
follows
immediately
from
Lemma
1.4,
assertion
that
VCN(
Γ
(iii),
(iv),
that,
by
replacing
Γ
by
Γ
÷
,
we
may
assume
without
loss
of
Σ
)
G
=
∅
generality
that
Γ
is
untangled.
Thus,
the
assertion
that
VCN(
Γ
follows
immediately
from
assertion
(vi).
This
completes
the
proof
of
Lemma
1.6.
Remark
1.6.1.
The
conclusion
of
Lemma
1.6,
(vi),
follows
for
an
arbitrary
[i.e.,
not
necessarily
solvable!]
finite
group
G
from
[ZM],
The-
orems
2.8,
2.10.
That
is
to
say,
the
proof
given
above
of
Lemma
1.6,
(vi),
may
be
regarded
as
an
alternative
proof
of
these
results
of
[ZM]
in
the
case
where
G
is
solvable.
In
this
context,
it
is
also
perhaps
of
inter-
est
to
observe
that,
by
considering
Lemma
1.6,
(vi),
in
the
case
where
Σ
=
Primes
and
“Γ”
is
taken
to
be
some
finite
connected
sub-semi-
disc
that
is
stabilized
by
the
action
of
G
[where
we
note
that
graph
of
Γ
disc
is
a
union
of
such
sub-semi-graphs],
one
one
verifies
easily
that
Γ
obtains
an
alternative
proof
of
the
classical
result
concerning
actions
of
finite
groups
on
trees
quoted
in
the
proofs
of
Lemma
1.6,
(iii);
[SemiAn],
Lemma
1.8,
(ii)
—
hence
also
alternative
proofs
of
Lemma
1.6,
(iii);
[SemiAn],
Lemma
1.8,
(ii)
—
in
the
case
where
the
finite
group
under
consideration
is
solvable.
Remark
1.6.2.
(i)
In
the
situation
of
Lemma
1.6,
if
G
is
isomorphic
to
Z/l
n
Z
for
some
prime
number
l
∈
Σ
and
a
positive
integer
n,
then
the
conclusion
of
the
resp’d
portion
of
Lemma
1.6,
(iv),
may
be
verified
by
the
following
easier
argument:
Since
[as
is
well-
known]
a
projective
limit
of
nonempty
finite
sets
is
nonempty,
Σ
)
G
=
∅,
it
suffices
to
verify
to
verify
the
assertion
that
VCN(
Γ
that
VCN(Γ
∗
)
G
=
∅
for
every
G-compatible
connected
finite
Σ
→
Γ.
Moreover,
one
verifies
im-
subcovering
Γ
∗
→
Γ
of
Γ
mediately
that
we
may
assume
without
loss
of
generality
that
24
YUICHIRO
HOSHI
AND
SHINICHI
MOCHIZUKI
Γ
∗
=
Γ.
Next,
let
us
observe
that
it
follows
immediately
from
Lemma
1.4,
(iv),
that,
by
replacing
Γ
by
Γ
÷
,
we
may
assume
without
loss
of
generality
that
G
acts
on
Γ
without
inversion.
def
Write
H
⊆
G
for
the
unique
subgroup
such
that
Q
=
G/H
def
is
of
order
l;
Γ
Q
=
Γ/H
for
the
“quotient
semi-graph”,
i.e.,
the
semi-graph
whose
vertices,
edges,
and
branches
are,
re-
spectively,
the
H-orbits
of
the
vertices,
edges,
and
branches
of
Γ
[cf.
the
fact
that
G
acts
on
Γ
without
inversion].
Then
one
verifies
immediately
that
the
natural
morphism
of
semi-graphs
Γ
Γ
Q
determines
an
outer
homomorphism
Σ
Π
Σ
Γ//G
−→
Π
Γ
Q
//Q
[cf.
the
notation
of
the
statement
of
Lemma
1.6].
Now
since
Π
Σ
Γ
Q
is
a
free
pro-Σ
group,
hence
torsion-free,
it
follows
that
the
restriction
s(H)
→
Π
Σ
Γ
Q
//Q
[which
clearly
factors
through
Σ
Σ
Σ
Π
Γ
Q
⊆
Π
Γ
Q
//Q
]
of
the
outer
homomorphism
Π
Σ
Γ//G
→
Π
Γ
Q
//Q
to
s(H)
⊆
Π
Σ
Γ//G
is
trivial,
hence
that
s
determines
a
section
Σ
s
Q
:
Q
→
Π
Γ
Q
//Q
of
the
natural
surjection
Π
Σ
Γ
Q
//Q
Q.
In
particular,
by
applying
Lemma
1.6,
(ii),
we
thus
conclude
that
VCN(Γ
Q
)
Q
=
∅.
Let
z
Q
∈
VCN(Γ
Q
)
Q
,
z
∈
VCN(Γ)
a
lifting
of
z
Q
,
and
g
∈
G
a
generator
of
G.
Then
since
Q
fixes
z
Q
,
it
follows
that
z
g
=
z
h
,
for
some
h
∈
H,
hence
that
z
is
fixed
by
g
·
h
−1
∈
G.
On
the
other
hand,
since
g
·
h
−1
generates
G,
we
thus
conclude
that
z
is
fixed
by
G,
i.e.,
that
VCN(Γ
∗
)
G
=
∅,
as
desired.
(ii)
The
proof
of
Lemma
1.6,
(ii),
as
well
as
the
argument
of
(i)
above,
is
essentially
the
same
as
the
argument
applied
in
[AbsCsp]
to
prove
[AbsCsp],
Lemma
2.1,
(iii).
Remark
1.6.3.
In
the
respective
situations
of
Lemma
1.6,
(iii),
(vi),
Σ
)
G
are
nec-
disc
)
G
and
the
sub-pro-semi-graph
(
Γ
the
sub-semi-graph
(
Γ
essarily
connected
[cf.
Lemma
1.6,
(iii),
(vi)].
On
the
other
hand,
Γ
G
is
not,
in
general,
connected.
This
phenomenon
may
be
seen
in
the
follow-
disc
is
the
graph
given
by
ing
example:
Suppose
that
2
∈
Σ,
and
that
Γ
the
integral
points
of
the
real
line
R,
i.e.,
the
vertices
are
given
by
the
elements
of
Z
⊆
R,
and
the
edges
are
given
by
the
closed
line
segments
joining
adjacent
elements
of
Z.
For
N
=
2M
a
positive
even
integer,
disc
by
the
evident
action
of
N
∈
Z
on
write
Γ
N
for
the
quotient
of
Γ
disc
via
translations.
Thus,
we
have
a
diagram
of
natural
covering
Γ
maps
disc
−→
Γ
N
−→
Γ
def
Γ
=
Γ
2
,
COMBINATORIAL
ANABELIAN
TOPICS
IV
25
and
the
group
G
=
Z/2Z
acts
equivariantly
on
this
diagram
via
mul-
tiplication
by
±1.
Here,
we
observe
that
since
N
is
even,
one
verifies
immediately
that
G
acts
on
Γ
N
without
inversion.
Then
one
computes
easily
that
disc
)
G
=
{0},
Γ
G
=
M
Z/N
Z.
(
Γ
N
Σ
G
In
particular,
the
pro-semi-graph
(
Γ
)
corresponds
to
the
inverse
limit
lim
M
Z/N
Z,
←−
hence
consists
of
a
single
pro-vertex
and
no
pro-edge
and,
in
particular,
is
nonempty
and
connected.
On
the
other
hand,
each
Γ
G
N
consists
of
precisely
two
vertices
and
no
edges,
hence
is
not
connected.
Definition
1.7.
Let
G
be
a
profinite
group
and
ρ
:
G
→
Aut(G)
a
continuous
homomorphism.
(i)
We
shall
say
that
ρ
is
of
ENN-type
[where
the
“ENN”
stands
for
“extended
NN”]
(respectively,
of
EPIPSC-type
[where
the
“EPIPSC”
stands
for
“extended
PIPSC”])
if
there
exists
a
nor-
mal
closed
subgroup
I
G
⊆
G
of
G
such
that,
for
every
open
ρ
subgroup
J
⊆
I
G
of
I
G
,
the
composite
J
→
G
→
Aut(G)
factors
as
a
composite
J
J
Σ-ab-free
→
Aut(G)
[cf.
the
dis-
cussion
entitled
“Groups”
in
§0],
where
the
second
arrow
is
of
NN-type
[cf.
[NodNon],
Definition
2.4,
(iii)]
(respectively,
of
PIPSC-type
[cf.
[CbTpIII],
Definition
1.3]).
In
this
situation,
we
shall
refer
to
I
G
as
a
conducting
subgroup.
Suppose
that
ρ
is
of
ENN-type
for
some
conducting
subgroup
I
G
⊆
G.
Then
we
shall
say
that
ρ
is
verticially
quasi-split
if
there
exists
an
open
subgroup
H
⊆
G
that
acts
as
the
identity
[i.e.,
relative
to
the
action
induced
by
ρ]
on
the
underlying
semi-graph
G
of
G
and,
moreover,
for
every
v
∈
Vert(G),
satisfies
the
following
condition:
the
extension
of
profinite
groups
[cf.
the
discussion
entitled
“Topological
groups”
in
[CbTpI],
§0]
out
1
−→
Π
v
−→
Π
v
H
−→
H
−→
1
—
where
Π
v
⊆
Π
G
is
a
verticial
subgroup
associated
to
v
∈
Vert(G)
—
associated
to
the
outer
action
of
H
on
Π
v
de-
termined
by
ρ
[cf.
[CmbGC],
Proposition
1.2,
(ii);
[CbTpI],
out
Lemma
2.12]
admits
a
splitting
s
v
:
H
→
Π
v
H
such
that
the
image
of
the
restriction
of
s
v
to
I
G
∩
H
commutes
with
Π
v
.
(ii)
Let
l
∈
Σ.
Then
we
shall
say
that
ρ
is
l-cyclotomically
full
if
χ
G
ρ
Σ
)
×
Z
×
[cf.
the
image
of
the
composite
G
→
Aut(G)
→
(
Z
l
[CbTpI],
Definition
3.8,
(ii)]
is
open.
26
YUICHIRO
HOSHI
AND
SHINICHI
MOCHIZUKI
Remark
1.7.1.
It
follows
immediately
from
[CbTpIII],
Remark
1.6.2,
that
the
following
implication
holds:
EPIPSC-type
=⇒
ENN-type.
Lemma
1.8
(Outer
representations
induced
on
pro-l
comple-
tions).
Let
G
be
a
profinite
group
and
ρ
:
G
→
Aut(G)
a
continu-
ous
homomorphism
of
ENN-type
(respectively,
of
EPIPSC-type)
for
a
conducting
subgroup
I
G
⊆
G
[cf.
Definition
1.7,
(i)].
For
l
∈
Σ,
write
G
{l}
for
the
semi-graph
of
anabelioids
of
pro-{l}
PSC-type
obtained
by
forming
the
pro-l
completion
of
G
[cf.
[SemiAn],
Defini-
ρ
tion
2.9,
(ii)].
Then
the
composite
G
→
Aut(G)
→
Aut(G
{l}
)
is
of
ENN-type
(respectively,
of
EPIPSC-type)
for
some
conducting
subgroup
⊆
G,
which
may
be
taken
to
be
a
normal
open
subgroup
of
I
G
.
Proof.
This
follows
immediately
from
the
various
definitions
involved
[cf.
also
[CbTpI],
Theorem
4.8,
(iv);
[CbTpI],
Corollary
5.9,
(ii),
(iii)].
Definition
1.9.
Let
z
∈
VCN(G).
If
z
∈
Vert(G)
(respectively,
z
∈
Edge(G)),
then
we
shall
refer
to
a
verticial
(respectively,
an
edge-like)
subgroup
of
Π
tp
G
associated
to
z
[cf.
[SemiAn],
Theorem
3.7,
(i),
(iii)]
as
a
VCN-subgroup
of
Π
tp
∈
VCN(
G
tp
),
we
shall
G
associated
to
z.
For
z
tp
also
speak
of
VCN-subgroups
of
Π
G
associated
to
z
.
Definition
1.10.
(i)
Let
Γ
be
a
semi-graph
and
v
∈
Vert(Γ).
Then
we
shall
write
V
δ≤1
(v)
⊆
Vert(Γ)
for
the
subset
consisting
of
w
∈
Vert(Γ)
such
that
either
w
=
v
or
N
(v)
∩
N
(w)
=
∅.
Also,
we
shall
def
write
Star(v)
=
V
δ≤1
(v)
E(v)
⊆
VCN(Γ).
(ii)
Let
v
∈
Vert(G).
Then
we
shall
write
V
δ≤1
(v)
⊆
Vert(G),
Star(v)
⊆
VCN(G)
for
the
respective
subsets
of
(i)
applied
to
the
underlying
semi-graph
of
G.
Then
we
shall
write
V
δ≤1
(
(iii)
Let
v
∈
Vert(
G).
v
)
⊆
Vert(
G),
for
the
respective
projective
limits
of
the
Star(
v
)
⊆
VCN(
G)
subsets
of
(ii),
i.e.,
where
the
constructions
of
these
subsets
are
applied
to
the
images
of
v
in
the
connected
finite
etale
subcoverings
of
G
→
G.
COMBINATORIAL
ANABELIAN
TOPICS
IV
27
Lemma
1.11
(VCN-subgroups
and
sections).
Let
G
be
a
profinite
group,
ρ
:
G
→
Aut(G)
a
continuous
homomorphism,
z
∈
VCN(
G),
z
tp
∈
VCN(
G
tp
),
Π
z
⊆
Π
G
a
VCN-subgroup
of
Π
G
associated
to
z
∈
and
Π
z
tp
⊆
Π
tp
a
VCN-subgroup
of
Π
tp
associated
to
z
tp
VCN(
G),
G
G
def
out
out
def
tp
[cf.
Definition
1.9].
Write
Π
G
=
Π
G
G,
Π
tp
G
=
Π
G
G
[cf.
the
discussion
entitled
“Topological
groups”
in
[CbTpI],
§0].
Thus,
we
have
a
natural
commutative
diagram
1
−−−→
Π
tp
−−−→
Π
tp
−−−→
G
−−−→
1
⏐
G
⏐
G
⏐
⏐
1
−−−→
Π
G
−−−→
Π
G
−−−→
G
−−−→
1
—
where
the
horizontal
sequences
are
exact;
the
vertical
arrows
are
tp
outer
injections;
Π
tp
G
acts
naturally
on
G
;
Π
G
acts
naturally
on
G.
Then
the
following
hold:
(i)
It
holds
that
Π
z
=
N
Π
G
(Π
z
)
∩
Π
G
=
C
Π
G
(Π
z
)
∩
Π
G
,
def
D
z
=
N
Π
G
(Π
z
)
=
C
Π
G
(Π
z
)
=
N
Π
G
(D
z
)
=
C
Π
G
(D
z
),
tp
Π
z
tp
=
N
Π
tp
(Π
z
tp
)
∩
Π
tp
G
=
C
Π
tp
(Π
z
tp
)
∩
Π
G
,
G
G
def
D
z
tp
=
N
Π
tp
(Π
z
tp
)
=
C
Π
tp
(Π
z
tp
)
=
N
Π
tp
(D
z
tp
)
=
C
Π
tp
(D
z
tp
).
G
G
G
G
(ii)
Suppose
that
ρ
is
of
ENN-type
for
a
conducting
subgroup
I
G
⊆
G
[cf.
Definition
1.7,
(i)].
Let
S
be
a
nonempty
subset
and
s
:
G
→
Π
G
a
section
of
the
surjection
Π
G
of
VCN(
G)
G
such
that,
for
each
y
∈
S,
it
holds
that
s(I
G
)
≺
D
y
[cf.
the
discussion
entitled
“Groups”
in
§0].
Then
there
exists
an
such
that
S
⊆
Star(
element
v
∈
Vert(
G)
v
)
[cf.
Definition
1.10,
(iii)].
(iii)
Suppose
that
ρ
is
of
ENN-type
for
a
conducting
subgroup
I
G
⊆
G.
Let
s
:
G
→
Π
G
be
a
section
of
the
surjection
Π
G
G
such
that
s(I
G
)
≺
D
z
[cf.
the
discussion
entitled
”Groups”
in
def
§0].
Write
G
s
=
C
Π
G
(s(I
G
)).
Then
there
exists
an
element
such
that
s(G)
⊆
G
s
⊆
D
z
.
z
∈
VCN(
G)
(iv)
Suppose
that
ρ
is
of
ENN-type
for
a
conducting
subgroup
tp
I
G
⊆
G.
Let
s
:
G
→
Π
tp
G
be
a
section
of
the
surjection
Π
G
G
such
that
s(I
G
)
≺
D
z
tp
[cf.
the
discussion
entitled
”Groups”
in
def
§0].
Write
G
s
=
C
Π
tp
(s(I
G
)).
Then
there
exists
an
element
G
(
z
)
tp
∈
VCN(
G
tp
)
such
that
s(G)
⊆
G
s
⊆
D
(
z
)
tp
.
In
par-
ticular,
G
s
is
contained
in
a
profinite
subgroup
of
Π
tp
G
[cf.
(i)].
28
YUICHIRO
HOSHI
AND
SHINICHI
MOCHIZUKI
Proof.
First,
we
verify
assertion
(i).
The
two
equalities
of
the
first
(respectively,
third)
line
of
the
display
and
the
first
“=”
of
the
sec-
ond
(respectively,
fourth)
line
of
the
display
follow
immediately
from
[CmbGC],
Proposition
1.2,
(i),
(ii)
(respectively,
[CmbGC],
Proposi-
tion
1.2,
(i),
(ii),
together
with
the
injection
reviewed
at
the
beginning
of
the
present
§1).
Thus,
the
second
and
third
“=”
of
the
second
(respectively,
fourth)
line
of
the
display
follow
immediately
from
the
chain
of
inclusions
D
z
⊆
N
Π
G
(D
z
)
⊆
C
Π
G
(D
z
)
⊆
C
Π
G
(D
z
∩
Π
G
)
=
C
Π
G
(Π
z
)
=
D
z
(respectively,
D
z
tp
⊆
N
Π
tp
(D
z
tp
)
⊆
C
Π
tp
(D
z
tp
)
⊆
C
Π
tp
(D
z
tp
∩Π
tp
G
)
=
C
Π
tp
(Π
z
tp
)
=
D
z
tp
)
G
G
G
G
—
where
the
third
“⊆”
follows
immediately
from
[CbTpII],
Lemma
3.9,
(i)
(respectively,
the
[easily
verified]
tempered
version
of
[CbTpII],
Lemma
3.9,
(i)).
This
completes
the
proof
of
assertion
(i).
Next,
we
verify
assertion
(ii).
Let
us
first
observe
that
it
follows
from
the
definition
of
the
term
“ENN-type”
that
the
restriction
of
ρ
to
I
G
⊆
G
factors
through
the
quotient
I
G
I
G
Σ-ab-free
[cf.
the
discussion
def
out
def
entitled
“Groups”
in
§0].
Write
Π
I
G
=
Π
G
I
G
and
Π
I
G
Σ-ab-free
=
out
Π
G
I
G
Σ-ab-free
.
Thus,
we
have
a
commutative
diagram
1
−−−→
Π
G
−−−→
Π
G
⏐
⏐
−−−→
G
⏐
⏐
−−−→
1
1
−−−→
Π
G
−−−→
Π
I
G
⏐
⏐
−−−→
I
G
⏐
⏐
−−−→
1
1
−−−→
Π
G
−−−→
Π
I
G
Σ-ab-free
−−−→
I
G
Σ-ab-free
−−−→
1
—
where
the
horizontal
sequences
are
exact,
the
upper
vertical
arrows
are
injective,
the
lower
vertical
arrows
are
surjective,
and
the
two
right-
hand
squares
are
cartesian.
Next,
let
us
observe
that
we
may
assume
without
loss
of
generality
that
S
is
equal
to
the
set
of
all
y
∈
VCN(
G)
such
that
s(I
G
)
≺
D
y
.
Now
since
s(I
G
)
≺
D
y
=
C
Π
G
(Π
y
)
[cf.
assertion
(i)]
for
every
y
∈
S,
it
holds
that,
for
each
y
∈
S,
some
open
subgroup
s
of
the
image
J
⊆
Π
I
G
Σ-ab-free
of
I
G
→
Π
I
G
Π
I
G
Σ-ab-free
is
contained
in
C
Π
I
Σ-ab-free
(Π
y
).
In
particular,
it
follows
from
[NodNon],
Propositions
G
3.8,
(i);
3.9,
(i),
(ii),
(iii),
that
•
every
pair
of
edges
∈
S
abut
to
a
common
vertex
∈
S;
•
the
distance
between
any
two
vertices
∈
S
is
≤
2
[cf.
Defini-
tion
1.1,
(iii)],
and
the
edges
“e
1
,
.
.
.
,
e
n
”
and
vertices
“v
0
,
.
.
.
,
v
n
”
of
loc.
cit.
may
be
taken
to
be
∈
S;
•
if
e
∈
S
is
an
edge,
then
V(
e
)
⊆
S.
COMBINATORIAL
ANABELIAN
TOPICS
IV
29
It
is
now
a
matter
of
elementary
combinatorial
graph
theory
[cf.
also
[NodNon],
Lemma
1.8]
to
conclude
that
S
⊆
Star(
v
)
for
some
v
∈
Vert(
G),
as
desired.
This
completes
the
proof
of
assertion
(ii).
Next,
we
verify
assertion
(iii).
Since
s(I
G
)
≺
D
z
,
the
action
of
some
open
subgroup
of
I
G
on
G
determined
by
s|
I
G
fixes
z
∈
VCN(
G).
Thus,
it
follows
from
the
definition
of
G
s
that,
if,
for
γ
∈
G
s
,
we
write
for
the
image
of
z
by
the
action
of
γ
∈
G
s
,
then
the
action
z
γ
∈
VCN(
G)
i.e.,
s(I
G
)
≺
D
z
γ
of
some
open
subgroup
of
I
G
on
G
fixes
z
γ
∈
VCN(
G),
for
every
γ
∈
G
s
.
Then
it
follows
from
assertion
(ii)
Now
suppose
that
z
∈
Edge(
G).
such
that
{
z
γ
|
γ
∈
G
s
}
⊆
E(
v
).
that
there
exists
a
vertex
v
∈
Vert(
G)
γ
Now
if
{
z
|
γ
∈
G
s
}
=
1,
then
it
is
immediate
that
G
s
⊆
D
z
.
On
the
other
hand,
if
{
z
γ
|
γ
∈
G
s
}
≥
2,
then
one
verifies
immediately
from
the
various
definitions
involved
[cf.
also
[NodNon],
Lemma
1.8]
that
the
which
thus
implies
that
G
s
⊆
D
v
.
This
action
of
G
s
fixes
v
∈
Vert(
G),
completes
the
proof
of
assertion
(iii)
in
the
case
where
z
∈
Edge(
G).
Then
it
follows
from
assertion
(ii)
Next,
suppose
that
z
∈
Vert(
G).
such
that
that
the
set
S
δ
of
vertices
v
∈
Vert(
G)
def
•
S
z
=
{
z
γ
|
γ
∈
G
s
}
⊆
V
δ≤1
(
v
);
•
any
edge
∈
Edge(
G)
that
abuts
to
two
distinct
elements
of
S
z
[hence
is
fixed
by
the
action,
determined
by
s|
I
G
,
of
some
open
subgroup
of
I
G
—
cf.
[NodNon],
Proposition
3.9,
(ii)]
necessarily
abuts
to
v
then
G
s
⊆
D
y
.
is
nonempty.
If
the
action
of
G
s
fixes
some
y
∈
VCN(
G),
Thus,
we
may
assume
without
loss
of
generality
that
the
action
of
G
s
In
particular,
it
follows
that
the
does
not
fix
any
element
of
VCN(
G).
[nonempty!]
sets
S
z
and
S
δ
—
both
of
which
are
clearly
preserved
by
the
action
of
G
s
—
are
of
cardinality
≥
2.
In
a
similar
vein,
S
δ
\
(S
δ
∩
S
z
)
is
either
empty
or
of
cardinality
≥
2.
Moreover,
the
latter
case
contradicts
[NodNon],
Lemma
1.8.
Thus,
we
conclude
that
S
δ
⊆
S
z
,
which,
by
the
definition
of
S
z
and
S
δ
,
implies
that
S
δ
=
S
z
,
i.e.,
that,
for
any
two
distinct
z
1
,
z
2
∈
S
z
,
there
exists
a
[unique,
by
[NodNon],
Lemma
1.8]
such
that
V(
e
∈
Edge(
G)
e
)
=
{
z
1
,
z
2
}.
But,
in
light
of
the
definition
contains
an
of
S
δ
,
this
implies
that
S
z
=
2,
and
hence
that
Edge(
G)
element
fixed
by
the
action
of
G
s
—
a
contradiction!
This
completes
hence
also
the
proof
of
assertion
(iii)
in
the
case
where
z
∈
Vert(
G),
of
assertion
(iii).
Assertion
(iv)
follows
immediately
from
a
similar
argument
to
the
argument
applied
in
the
proof
of
assertion
(iii).
This
completes
the
proof
of
Lemma
1.11.
30
YUICHIRO
HOSHI
AND
SHINICHI
MOCHIZUKI
Lemma
1.12
(Triviality
via
passage
to
abelianizations).
Let
G
and
J
be
profinite
groups
and
φ
:
J
→
G
a
continuous
homomorphism.
Then
the
following
hold:
(i)
Let
γ
∈
G
be
such
that,
for
every
open
subgroup
H
⊆
G
of
G
that
contains
γ,
the
image
of
γ
in
H
ab
is
trivial.
Then
γ
is
trivial.
(ii)
Suppose
that,
for
every
open
subgroup
H
⊆
G
of
G,
the
com-
φ
posite
φ
−1
(H)
→
H
H
ab
is
trivial.
Then
φ
is
trivial.
Proof.
First,
we
verify
assertion
(i).
Assume
that
γ
is
nontrivial.
Then
it
is
immediate
that
there
exists
a
normal
open
subgroup
N
⊆
G
of
G
such
that
γ
∈
N
.
Write
H
⊆
G
for
the
closed
subgroup
of
G
topologically
generated
by
N
and
γ.
Then
the
image
of
γ
in
the
abelian
quotient
H
H/N
is
nontrivial.
This
completes
the
proof
of
assertion
(i).
Assertion
(ii)
follows
immediately
from
assertion
(i).
This
completes
the
proof
of
Lemma
1.12.
Theorem
1.13
(The
combinatorial
section
conjecture
for
outer
representations
of
ENN-type).
Let
Σ
be
a
nonempty
set
of
prime
numbers,
G
a
semi-graph
of
anabelioids
of
pro-Σ
PSC-type,
G
a
profi-
nite
group,
and
ρ
:
G
→
Aut(G)
a
continuous
homomorphism
that
is
of
ENN-type
for
a
conducting
subgroup
I
G
⊆
G
[cf.
Definition
1.7,
(i)].
Write
Π
G
for
the
[pro-Σ]
fundamental
group
of
G
and
Π
tp
G
for
the
tempered
fundamental
group
of
G
[cf.
[SemiAn],
Example
2.10;
the
dis-
cussion
preceding
[SemiAn],
Proposition
3.6].
[Thus,
we
have
a
natural
outer
injection
Π
tp
G
→
Π
G
—
cf.
[CbTpIII],
Lemma
3.2,
(i);
the
proof
of
def
out
[CbTpIII],
Proposition
3.3,
(i),
(ii).]
Write
Π
G
=
Π
G
G
[cf.
the
dis-
def
out
tp
cussion
entitled
“Topological
groups”
in
[CbTpI],
§0];
Π
tp
G
=
Π
G
G;
G
→
G,
G
tp
→
G
for
the
universal
pro-Σ
and
pro-tempered
coverings
of
G
corresponding
to
Π
G
,
Π
tp
G
;
VCN(−)
for
the
set
of
vertices,
cusps,
and
nodes
of
the
underlying
[pro-]semi-graph
of
a
[pro-]semi-graph
of
an-
abelioids
[cf.
Definition
1.1,
(i)].
Thus,
we
have
a
natural
commutative
diagram
1
−−−→
Π
tp
−−−→
Π
tp
−−−→
G
−−−→
1
⏐
G
⏐
G
⏐
⏐
1
−−−→
Π
G
−−−→
Π
G
−−−→
G
−−−→
1
—
where
the
horizontal
sequences
are
exact;
the
vertical
arrows
are
tp
outer
injections;
Π
tp
G
acts
naturally
on
G
;
Π
G
acts
naturally
on
G.
Then
the
following
hold:
(i)
Suppose
that
ρ
is
l-cyclotomically
full
[cf.
Definition
1.7,
(ii)]
for
some
l
∈
Σ.
Let
s
:
G
→
Π
G
be
a
continuous
section
of
the
natural
surjection
Π
G
G.
Then,
relative
to
the
action
of
COMBINATORIAL
ANABELIAN
TOPICS
IV
31
via
conjugation
of
VCN-subgroups,
the
image
Π
G
on
VCN(
G)
of
s
stabilizes
some
element
of
VCN(
G).
tp
(ii)
Let
s
:
G
→
Π
G
be
a
continuous
section
of
the
natural
surjec-
tp
tp
tion
Π
tp
G
G.
Then,
relative
to
the
action
of
Π
G
on
VCN(
G
)
via
conjugation
of
VCN-subgroups
[cf.
Definition
1.9],
the
im-
age
of
s
stabilizes
some
element
of
VCN(
G
tp
).
(iii)
Write
Sect(Π
G
/G)
for
the
set
of
Π
G
-conjugacy
classes
of
con-
tinuous
sections
of
the
natural
surjective
homomorphism
Π
G
tp
G
and
Sect(Π
tp
G
/G)
for
the
set
of
Π
G
-conjugacy
classes
of
continuous
sections
of
the
natural
surjective
homomorphism
Π
tp
G
G.
Then
the
natural
map
Sect(Π
tp
G
/G)
−→
Sect(Π
G
/G)
is
injective.
If,
moreover,
ρ
is
l-cyclotomically
full
for
some
l
∈
Σ,
then
this
map
is
bijective.
Proof.
First,
we
verify
assertion
(i).
Let
us
first
observe
that
by
replac-
ing
I
G
by
a
suitable
open
subgroup
of
I
G
and
G
by
the
pro-l
completion
of
the
finite
étale
covering
of
G
determined
by
a
varying
normal
open
subgroup
H
⊆
Π
G
such
that
s(G)
⊆
H
[cf.
Lemma
1.8;
[CbTpIII],
Lemma
1.5],
it
follows
immediately
from
the
well-known
fact
that
a
projective
limit
of
nonempty
finite
sets
is
nonempty
that
we
may
as-
sume
without
loss
of
generality
that
Σ
=
{l}.
Next,
let
us
observe
that
we
may
assume
without
loss
of
generality
that
G
has
at
least
one
node.
In
particular,
it
follows
immediately
from
Lemma
1.11,
(iii),
that,
to
verify
assertion
(i),
by
replacing
Π
G
by
a
suitable
open
subgroup
of
Π
G
,
we
may
assume
without
loss
of
generality
—
i.e.,
by
arguing
as
in
the
discussion
entitled
“Curves”
in
[AbsTpII],
§0
—
that
the
pro-l
completion
Π
G
of
the
topological
fundamental
group
of
the
underlying
semi-graph
G
of
G
is
a
free
pro-l
group
of
rank
≥
2,
hence,
in
particular,
center-free.
Then
we
claim
that
the
following
assertion
holds:
Claim
1.13.A:
For
every
connected
finite
étale
Galois
subcovering
H
→
G
of
G
→
G
that
determines
a
nor-
mal
open
subgroup
of
Π
G
,
the
action
of
I
G
on
H,
via
s,
fixes
an
element
of
VCN(H).
To
verify
Claim
1.13.A,
let
us
observe
that,
by
replacing
H
by
G
[cf.
[CbTpIII],
Lemma
1.5],
we
may
assume
without
loss
of
generality
that
H
=
G.
Next,
let
us
observe
that
since
the
underlying
semi-graph
G
of
G
is
finite,
the
continuous
action
of
G
on
G
factors
through
a
finite
def
quotient
G
Q,
i.e.,
by
a
normal
open
subgroup
of
G.
Write
Π
G//Q
=
out
Π
G
Q
[i.e.,
notation
which
is
well-defined
since
Π
G
is
center-free
—
cf.
the
discussion
entitled
“Topological
groups”
in
[CbTpI],
§0;
the
notational
conventions
of
Lemma
1.6,
in
the
case
where
“Σ”
is
taken
32
YUICHIRO
HOSHI
AND
SHINICHI
MOCHIZUKI
to
be
{l}].
Thus,
we
obtain
a
commutative
diagram
1
−−−→
Π
G
−−−→
Π
G
−−−→
G
−−−→
1
⏐
⏐
⏐
⏐
⏐
⏐
1
−−−→
Π
G
−−−→
Π
G//Q
−−−→
Q
−−−→
1
—
where
the
horizontal
sequences
are
exact,
and
the
vertical
arrows
are
surjective.
Write
I
G
I
Q
for
the
[finite]
quotient
of
I
G
determined
by
def
def
the
quotient
G
Q,
N
G
=
Ker(G
Q),
and
N
I
=
Ker(I
G
I
Q
).
Now
let
us
observe
that
(a)
since
Q
is
finite,
it
is
immediate
that
N
G
,
N
I
are
open
in
G,
I
G
,
respectively,
and,
moreover,
(b)
it
follows
from
the
definition
of
the
term
“ENN-type”
that,
by
replacing
G
Q
by
a
suitable
quotient
of
Q
if
necessary,
we
may
assume
without
loss
of
generality
that
the
quotient
{l}-ab-free
[cf.
the
I
G
I
Q
factors
through
the
quotient
I
G
I
G
discussion
entitled
“Groups”
in
§0],
hence
is
cyclic
of
order
a
power
of
l.
s
Next,
let
us
observe
that
the
composite
N
G
→
G
→
Π
G
Π
G//Q
determines
a
commutative
diagram
N
I
−→
N
G
|
|
↓
↓
Π
G
==
Π
G
—
where
the
upper
horizontal
arrow
is
the
natural
inclusion.
Now
we
claim
that
the
following
assertion
holds:
Claim
1.13.B:
The
left-hand
vertical
arrow
N
I
→
Π
G
of
the
above
diagram
is
the
trivial
homomorphism.
Indeed,
let
H
⊆
Π
G
be
an
open
subgroup
and
write
N
I,H
⊆
N
I
and
N
G,H
⊆
N
G
for
the
open
subgroups
obtained
by
forming
the
inverse
image
of
H
⊆
Π
G
via
the
vertical
arrows
of
the
above
commutative
diagram.
Thus,
N
G,H
normalizes
N
I,H
;
the
action
of
N
G,H
on
H
by
conjugation
induces
the
trivial
action
of
N
G,H
on
H
ab
.
Next,
let
us
observe
that
since
H
ab
is
a
free
Z
l
-module,
the
left-hand
vertical
arrow
{l}-ab-free
→
H
ab
of
under
consideration
determines
a
homomorphism
N
I,H
free
Z
l
-modules
of
finite
rank
[cf.
Definition
1.7,
(i)],
which
is
N
G,H
-
equivariant
[with
respect
to
the
actions
of
N
G,H
by
conjugation].
On
the
other
hand,
since
the
action
of
N
G,H
on
H
ab
is
trivial,
the
N
G,H
-
{l}-ab-free
equivariant
homomorphism
N
I,H
→
H
ab
factors
through
a
quo-
{l}-ab-free
tient
of
N
I,H
on
which
N
G,H
acts
trivially.
Thus,
since
ρ
is
l-
{l}-ab-free
cyclotomically
full,
and
N
G,H
acts
on
N
I,H
via
the
cyclotomic
character
[cf.
Definition
1.7,
(i);
[CbTpI],
Lemma
5.2,
(ii)],
we
con-
{l}-ab-free
clude
that
the
N
G,H
-equivariant
homomorphism
N
I,H
→
H
ab
is
COMBINATORIAL
ANABELIAN
TOPICS
IV
33
trivial.
In
particular,
Claim
1.13.B
follows
from
Lemma
1.12,
(ii).
This
completes
the
proof
of
Claim
1.13.B.
Next,
let
us
observe
that
it
follows
immediately
from
Claim
1.13.B
that
the
section
s
determines
a
section
of
the
natural
surjection
def
pr
2
Π
G//I
Q
=
Π
G//Q
×
Q
I
Q
I
Q
.
Thus,
it
follows
immediately
from
the
resp’d
portion
of
Lemma
1.6,
(iv),
together
with
the
observation
(b)
discussed
above
[cf.
also
Re-
mark
1.13.1
below],
that
Claim
1.13.A
holds.
This
completes
the
proof
of
Claim
1.13.A.
Now
by
allowing
the
subcovering
H
in
Claim
1.13.A
to
vary,
we
conclude
immediately
[from
the
well-known
fact
that
a
projective
limit
of
nonempty
finite
sets
is
nonempty]
that
s(I
G
)
stabilizes
some
element
Thus,
it
follows
from
Lemma
1.11,
(iii),
that
the
image
of
VCN(
G).
This
completes
the
proof
of
s(G)
stabilizes
some
element
of
VCN(
G).
assertion
(i).
Assertion
(ii)
follows,
by
applying
[NodNon],
Proposition
3.9,
(i),
from
a
similar
argument
to
the
argument
applied
to
prove
[SemiAn],
Theorems
3.7,
5.4.
That
is
to
say,
instead
of
considering
“subjoints”
[i.e.,
paths
of
length
2]
as
in
the
proof
of
[SemiAn],
Theorem
3.7,
the
content
of
[NodNon],
Proposition
3.9,
(i),
requires
us
to
consider
paths
of
length
3.
This
completes
the
proof
of
assertion
(ii).
Finally,
we
verify
assertion
(iii).
Let
s,
t
:
G
→
Π
tp
G
be
sections
of
tp
the
surjection
Π
G
G
such
that
there
exists
an
element
γ
∈
Π
G
such
s
that
the
composite
s
:
G
→
Π
tp
G
→
Π
G
is
the
conjugate
by
γ
∈
Π
G
of
t
tp
the
composite
t
:
G
→
Π
G
→
Π
G
.
Thus,
it
follows
from
assertion
(ii)
[applied
to
both
s
and
t]
that
there
exist
elements
y
,
z
∈
VCN(
G
tp
)
for
the
image
of
z
by
the
action
such
that
if
we
write
z
γ
∈
VCN(
G)
of
γ,
then
s
stabilizes
both
y
and
z
γ
.
In
particular,
we
conclude
from
Lemma
1.11,
(ii),
that
the
distance
between
y
and
z
γ
is
finite,
hence
that,
for
each
subcovering
H
→
G
of
G
tp
→
G
that
arises
from
an
and
z
γ
in
open
subgroup
of
Π
tp
G
,
the
distance
between
the
images
of
z
tp
H
is
finite,
which
implies
that
γ
∈
Π
G
.
This
completes
the
proof
of
the
injectivity
portion
of
assertion
(iii).
Since
[one
verifies
immediately
lies
in
the
Π
G
-orbit
of
an
element
of
that]
every
element
of
VCN(
G)
VCN(
G
tp
),
the
final
portion
of
assertion
(iii)
follows
immediately
from
assertion
(i).
This
completes
the
proof
of
Theorem
1.13.
Remark
1.13.1.
We
observe
in
passing,
with
regard
to
the
application
of
Lemma
1.6,
(iv),
in
the
proof
of
Theorem
1.13,
(i),
that,
in
fact,
Lemma
1.6,
(iv),
is
only
applied
in
the
case
where
the
group
“G”
of
Lemma
1.6
is
cyclic
and
of
order
a
power
of
l.
That
is
to
say,
we
only
34
YUICHIRO
HOSHI
AND
SHINICHI
MOCHIZUKI
apply
Lemma
1.6,
(iv),
in
the
case
that,
as
discussed
in
Remark
1.6.2,
(i),
admits
a
relatively
simple
proof.
Corollary
1.14
(A
combinatorial
version
of
the
Grothendieck
conjecture
for
outer
representations
of
ENN-type).
Let
Σ
be
a
nonempty
set
of
prime
numbers;
G,
H
semi-graphs
of
anabelioids
of
∼
pro-Σ
PSC-type;
G
G
,
G
H
profinite
groups;
β
:
G
G
→
G
H
a
continu-
ous
isomorphism;
ρ
G
:
G
G
→
Aut(G),
ρ
H
:
G
H
→
Aut(H)
continuous
homomorphisms
that
are
of
ENN-type
for
conducting
subgroups
I
G
G
⊆
G
G
,
I
G
H
⊆
G
H
[cf.
Definition
1.7,
(i)]
such
that
β(I
G
G
)
=
I
G
H
;
l
∈
Σ
a
prime
number
such
that
ρ
G
and
ρ
H
are
l-cyclotomically
full
[cf.
Definition
1.7,
(ii)].
Suppose
further
that
ρ
G
is
verticially
quasi-
split
[cf.
Definition
1.7,
(i)].
Write
Π
G
,
Π
H
for
the
[pro-Σ]
funda-
∼
mental
groups
of
G,
H,
respectively.
Let
α
:
Π
G
→
Π
H
be
a
continuous
isomorphism
such
that
the
diagram
G
G
ρ
G
−→
Aut(G)
−→
Out(Π
G
)
|
|
↓
β
↓
G
H
ρ
H
−→
Aut(H)
−→
Out(Π
H
)
—
where
the
right-hand
vertical
arrow
is
the
isomorphism
obtained
by
conjugating
by
α
—
commutes.
Then
α
is
graphic
[cf.
[CmbGC],
Definition
1.4,
(i)].
Proof.
First,
let
us
observe
that
by
[CmbGC],
Corollary
2.7,
(i),
it
follows
from
our
assumption
that
ρ
G
and
ρ
H
are
l-cyclotomically
full
∼
that
α
:
Π
G
→
Π
H
is
group-theoretically
cuspidal.
Thus,
by
applying
[CmbGC],
Proposition
1.5,
(ii);
[NodNon],
Lemma
1.14,
we
conclude
that
it
suffices
to
verify
that
α
is
group-theoretically
verticial
under
the
additional
assumption
that
G
and
H
are
noncuspidal.
Write
Π
G
G
,
Π
G
H
for
the
profinite
groups
“Π
G
”
[cf.
Theorem
1.13]
associated
to
ρ
G
,
ρ
H
.
Then
it
follows
immediately
from
our
assumption
that
ρ
G
is
verticially
quasi-split
that
we
may
assume,
after
possibly
replacing
G
G
and
G
H
by
corresponding
open
subgroups,
that
there
exists
a
section
s
G
:
G
G
→
Π
G
G
such
that
the
image
of
the
restriction
of
s
G
to
I
G
G
com-
mutes
with
some
verticial
subgroup
of
Π
G
.
In
particular,
s
G
satisfies
the
conditions
imposed
on
the
section
“s
:
G
→
Π
G
”
in
Lemma
1.11,
(ii),
for
some
nonempty
subset
“S”.
Moreover,
it
follows
from
The-
∼
orem
1.13,
(i),
that
the
isomorphism
Π
G
G
→
Π
G
H
determined
by
α
and
β
maps
s
G
to
a
section
s
H
:
G
H
→
Π
G
H
that
is
contained
in
the
normalizer
in
Π
G
H
of
a
VCN-subgroup
of
Π
H
.
In
particular,
after
possibly
replacing
G
G
and
G
H
by
corresponding
open
subgroups,
we
may
assume
[cf.
[CmbGC],
Proposition
1.2,
(ii);
[NodNon],
Remark
COMBINATORIAL
ANABELIAN
TOPICS
IV
35
2.7.1]
that
the
image
of
the
restriction
of
s
H
to
I
G
H
commutes
with
some
nontrivial
verticial
element
of
Π
H
[cf.
[CbTpII],
Definition
1.1].
Thus,
by
restricting
these
sections
s
G
,
s
H
to
the
respective
conducting
subgroups
and
forming
appropriate
centralizers
[cf.
[NodNon],
Lemma
3.6,
(i),
applied
to
the
restriction
of
s
G
to
I
G
G
],
we
conclude
from
the
assumption
that
β
is
compatible
with
the
respective
conducting
sub-
∼
groups
that
α
:
Π
G
→
Π
H
maps
some
nontrivial
verticial
element
of
Π
G
to
a
nontrivial
verticial
element
of
Π
H
.
In
particular,
it
follows
from
the
implication
(3)
⇒
(1)
of
[CbTpII],
Theorem
1.9,
(i),
that
α
is
group-theoretically
verticial,
as
desired.
Remark
1.14.1.
It
is
not
difficult
to
verify
that
the
assumption
in
the
statement
of
Corollary
1.14
that
β(I
G
G
)
=
I
G
H
cannot
be
omitted.
Indeed,
if
one
omits
this
assumption,
then
a
counterexample
to
the
graphicity
asserted
in
Corollary
1.14
may
be
obtained
as
follows:
Let
J
be
a
semi-graph
of
anabelioids
of
pro-Σ
PSC-type
and
e
G
,
e
H
distinct
nodes
of
J
.
Write
G
(respectively,
H)
for
the
semi-graph
of
anabe-
lioids
of
pro-Σ
PSC-type
J
Node(J
)\{e
G
}
(respectively,
J
Node(J
)\{e
H
}
)
obtained
by
deforming
the
nodes
of
J
that
are
=
e
G
(respectively,
=
e
H
)
[cf.
[CbTpI],
Definition
2.8];
I
G
G
(respectively,
I
G
H
)
for
the
[nec-
essarily
normal
—
cf.
[CbTpI],
Theorem
4.8,
(i),
(v)]
closed
subgroup
of
Aut
|{e
G
,e
H
}|
(J
)
[cf.
[CbTpI],
Definition
2.6,
(i)]
generated
by
the
profi-
nite
Dehn
twists
that
arise
from
the
direct
summand
of
the
direct
sum
decomposition
in
the
display
of
[CbTpI],
Theorem
4.8,
(iv),
labeled
by
e
G
(respectively,
e
H
).
Next,
let
G
G
=
G
H
be
a
closed
subgroup
of
Aut
|{e
G
,e
H
}|
(J
)
such
that
•
G
G
=
G
H
contains
both
I
G
G
and
I
G
H
,
•
the
natural
inclusion
G
G
=
G
H
→
Aut(J
)
is
l-cyclotomically
full
for
some
l
∈
Σ,
and,
moreover,
•
if
we
write
ρ
G
(respectively,
ρ
H
)
for
the
continuous
injection
G
G
→
Aut(G)
(respectively,
G
H
→
Aut(H))
obtained
by
forming
the
composite
of
the
natural
inclusion
G
G
=
G
H
→
Aut
|{e
G
,e
H
}|
(J
)
and
the
injection
Aut
|{e
G
,e
H
}|
(J
)
→
Aut(G)
(respectively,
Aut
|{e
G
,e
H
}|
(J
)
→
Aut(H))
[cf.
[CbTpI],
Propo-
sition
2.9,
(ii)],
then
ρ
G
is
verticially
quasi-split.
[Note
that
one
verifies
easily
the
existence
of
such
a
closed
subgroup
of
Aut
|{e
G
,e
H
}|
(J
)
by
considering,
for
instance,
a
homomorphism
G
G
=
G
H
→
Aut(J
)
of
EPIPSC-type
that
arises
from
a
suitable
stable
log
curve
—
cf.
also
Remark
1.7.1;
[CbTpI],
Lemma
5.4,
(ii);
[CbTpI],
Proposition
5.6,
(ii).]
Then
if
one
takes
the
“α”
of
Corollary
1.14
to
be
the
outer
isomorphism
determined
by
the
specialization
outer
isomorphisms
Φ
J
Node
(J
)\{e
G
}
,
Φ
J
Node
(J
)\{e
H
}
[cf.
[CbTpI],
Definition
2.10]
and
the
“β”
of
Corollary
1.14
to
be
the
identity
isomorphism,
then
36
YUICHIRO
HOSHI
AND
SHINICHI
MOCHIZUKI
one
verifies
immediately
from
[CbTpI],
Corollary
3.9,
(i),
and
[CbTpI],
Corollary
5.9,
(iii),
that
one
obtains
a
counterexample
as
desired.
Let
R
be
a
complete
discrete
valuation
ring
whose
residue
character-
istic
we
denote
by
p
[so
p
may
be
zero];
K
a
separable
closure
of
the
field
of
fractions
K
of
R;
X
log
a
stable
log
curve
[cf.
the
discussion
entitled
“Curves”
in
[CbTpI],
§0]
over
the
log
regular
log
scheme
Spec(R)
log
obtained
by
equipping
Spec(R)
with
the
log
structure
determined
by
the
maximal
ideal
m
R
⊆
R
of
R.
Suppose,
for
simplicity,
that
X
log
is
split,
i.e.,
that
the
nat-
ural
action
of
Gal(K/K)
on
the
dual
semi-graph
Γ
X
log
associated
to
def
the
geometric
special
fiber
of
X
log
is
trivial.
Write
X
log
=
X
log
×
R
K;
Vert(X
log
)
(respectively,
Cusp(X
log
);
Node(X
log
))
for
the
set
of
ver-
tices
(respectively,
open
edges;
closed
edges)
of
Γ
X
log
,
i.e.,
the
set
of
connected
components
of
the
complement
of
the
cusps
and
nodes
(respectively,
the
set
of
cusps;
the
set
of
nodes)
of
the
special
fiber
of
X
log
;
def
VCN(X
log
)
=
Vert(X
log
)
Cusp(X
log
)
Node(X
log
).
Before
proceeding,
we
recall
that
to
each
element
z
∈
VCN(X
log
),
one
may
associate,
in
a
way
that
is
functorial
with
respect
to
arbitrary
auto-
morphisms
of
the
log
scheme
X
log
,
a
discrete
valuation
that
dominates
R
on
the
residue
field
of
some
point
of
X,
which
is
closed
if
and
only
if
z
is
a
cusp.
Indeed,
this
is
immediate
if
z
is
a
vertex,
since
a
vertex
corresponds
to
a
prime
of
height
1
of
X
.
This
is
also
immediate
if
z
is
a
cusp,
since
the
residue
field
of
the
closed
point
of
X
that
corresponds
to
z
is
finite
over
[the
complete
discrete
valuation
field]
K,
which
implies
that
the
discrete
valuation
of
K
extends
uniquely
to
a
discrete
valuation
on
the
residue
field
of
a
cusp.
Now
suppose
that
z
is
a
node
that
is
locally
defined
by
an
equation
of
the
form
s
1
s
2
−
a,
for
some
a
∈
m
R
[cf.,
e.g.,
the
discussion
of
[CbTpI],
Definition
5.3,
(ii)].
By
descent,
we
may
assume
without
loss
of
generality
that
a
admits
a
square
root
b
in
R.
Then
one
associates
to
z
the
discrete
valuation
determined
by
the
exceptional
divisor
of
the
blow-up
of
X
at
the
locus
(s
1
,
s
2
,
b).
[One
verifies
immediately
that
this
construction
is
compatible
with
arbitrary
automorphisms
of
X
log
.]
Corollary
1.15
(Fixed
points
associated
to
Galois
sections).
Let
Σ
be
a
set
of
prime
numbers;
Σ
†
⊆
Σ
a
subset;
l
∈
Σ
†
;
R
a
complete
COMBINATORIAL
ANABELIAN
TOPICS
IV
37
discrete
valuation
ring
of
residue
characteristic
p
∈
Σ
†
[so
p
may
be
zero];
K
a
separable
closure
of
the
field
of
fractions
K
of
R;
X
log
a
stable
log
curve
[cf.
the
discussion
entitled
“Curves”
in
[CbTpI],
§0]
over
the
log
regular
log
scheme
Spec(R)
log
obtained
by
equipping
Spec(R)
with
the
log
structure
determined
by
the
maximal
ideal
of
R.
def
Write
G
K
=
Gal(K/K)
for
the
absolute
Galois
group
of
K;
I
K
⊆
G
K
def
log
def
for
the
inertia
subgroup
of
G
K
;
X
log
=
X
log
×
R
K;
X
K
=
X
log
×
R
K;
Δ
X
log
log
for
the
pro-Σ
log
fundamental
group
of
X
K
[i.e.,
the
maximal
pro-Σ
log
quotient
of
the
log
fundamental
group
of
X
K
];
Π
X
log
for
the
geometrically
pro-Σ
log
fundamental
group
of
X
log
[i.e.,
the
quotient
of
the
log
fundamental
group
of
X
log
by
the
kernel
of
the
natural
log
surjection
from
the
log
fundamental
group
of
X
K
onto
Δ
X
log
].
Thus,
we
have
a
natural
exact
sequence
of
profinite
groups
1
−→
Δ
X
log
−→
Π
X
log
−→
G
K
−→
1.
log
→
X
log
for
the
profinite
log
étale
covering
of
X
log
corre-
Write
X
sponding
to
Π
X
log
.
If
Y
log
→
X
log
is
a
finite
connected
subcovering
of
log
→
X
log
that
admits
a
stable
model
Y
log
over
the
normalization
R
Y
X
of
R
in
Y
,
then
let
us
write
Γ
Y
log
for
the
dual
semi-graph
determined
by
the
geometric
special
fiber
of
Y
log
over
R
Y
;
Vert(Y
log
)
(respectively,
Cusp(Y
log
);
Node(Y
log
))
for
the
set
of
vertices
(respectively,
open
edges;
closed
edges)
of
Γ
Y
log
,
i.e.,
the
set
of
connected
components
of
the
com-
plement
of
the
cusps
and
nodes
(respectively,
the
set
of
cusps;
the
set
of
nodes)
of
the
geometric
special
fiber
of
Y
log
over
R
Y
;
def
Edge(Y
log
)
=
Cusp(Y
log
)
Node(Y
log
);
def
VCN(Y
log
)
=
Vert(Y
log
)
Edge(Y
log
);
log
)
=
lim
VCN(Y
log
)
VCN(
X
←−
—
where
the
projective
limit
is
over
all
finite
connected
subcoverings
log
→
X
log
as
above,
and,
moreover,
for
each
finite
Y
log
→
X
log
of
X
log
→
X
log
that
admits
a
stable
connected
subcovering
Y
1
log
→
X
log
of
X
log
model
Y
1
over
the
normalization
of
R
in
Y
1
,
the
transition
map
for
log
→
Y
1
log
that
admits
a
finite
connected
subcovering
Y
2
log
→
Y
1
log
of
X
a
stable
model
Y
2
log
over
the
normalization
of
R
in
Y
2
is
defined,
for
z
∈
VCN(Y
2
log
),
as
follows:
def
38
YUICHIRO
HOSHI
AND
SHINICHI
MOCHIZUKI
•
If
the
connected
component/cusp/node
corresponding
to
z
maps,
via
the
extension
Y
2
log
→
Y
1
log
of
Y
2
log
→
Y
1
log
[cf.,
e.g.,
[ExtFam],
Theorem
C],
to
a
cusp
or
node
of
the
geometric
special
fiber
of
Y
1
,
then
the
image
of
z
∈
VCN(Y
2
log
)
in
VCN(Y
1
log
)
is
defined
to
be
the
element
of
Edge(Y
1
log
)
corresponding
to
the
cusp
or
node.
•
If
the
generic
point
of
the
connected
component/cusp/node
cor-
responding
to
z
maps,
via
the
extension
Y
2
log
→
Y
1
log
of
Y
2
log
→
Y
1
log
,
to
a
point
of
the
geometric
special
fiber
of
Y
1
that
is
neither
a
cusp
nor
node,
then
the
image
of
z
∈
VCN(Y
2
log
)
in
VCN(Y
1
log
)
is
defined
to
be
the
element
of
Vert(Y
1
log
)
corre-
sponding
to
the
connected
component
on
which
the
point
lies.
log
),
and
Y
log
→
X
log
is
a
finite
connected
subcovering
If
z
∈
VCN(
X
log
→
X
log
that
admits
a
stable
model
Y
log
over
the
normalization
of
X
of
R
in
Y
,
then
let
us
write
z
(Y
log
)
∈
VCN(Y
log
)
for
the
element
of
VCN(Y
log
)
determined
by
z
.
Let
H
⊆
G
K
be
a
closed
subgroup
such
that
the
image
of
def
I
H
=
H
∩
I
K
⊆
I
K
Σ
†
Σ
†
via
the
natural
surjection
I
K
I
K
to
the
pro-Σ
†
completion
I
K
of
I
K
†
Σ
is
an
open
subgroup
of
I
K
and
s
:
H
−→
Π
X
log
a
section
of
the
restriction
to
H
⊆
G
K
of
the
above
exact
sequence
1
→
Δ
X
log
→
Π
X
log
→
G
K
→
1.
Then
the
following
hold:
(i)
If
we
write
Δ
†
X
log
for
the
maximal
pro-Σ
†
quotient
of
Δ
X
log
and
regard,
via
the
specialization
outer
isomorphism
with
respect
to
X
log
,
the
pro-Σ
†
group
Δ
†
X
log
as
the
[pro-Σ
†
]
fundamental
group
of
the
semi-graph
of
anabelioids
of
pro-Σ
†
PSC-type
de-
termined
by
the
geometric
special
fiber
of
the
stable
model
X
log
[cf.
[CmbGC],
Example
2.5],
then
the
natural
outer
Galois
ac-
tion
H
−→
Out(Δ
†
X
log
)
determined
by
the
above
exact
sequence
is
of
EPIPSC-type
for
the
conducting
subgroup
I
H
⊆
H
[cf.
Definition
1.7,
(i)].
If,
moreover,
H
is
l-cyclotomically
full,
i.e.,
the
image
of
H
⊆
G
K
via
the
l-adic
cyclotomic
character
on
G
K
is
open,
then
the
above
outer
Galois
action
is
l-cyclotomically
full
[cf.
Definition
1.7,
(ii)].
(ii)
Suppose
that
the
residue
field
of
R
is
countable.
Let
z
∈
log
)
and
S
=
{Y
log
→
X
log
}
a
cofinal
system
consisting
VCN(
X
log
→
X
log
such
of
finite
Galois
subcoverings
Y
log
→
X
log
of
X
that
Y
log
admits
a
split
stable
model
over
the
normalization
R
Y
of
R
in
Y
.
Then
there
exist
a
valuation
v
z
on
the
residue
COMBINATORIAL
ANABELIAN
TOPICS
IV
39
of
X
log
[i.e.,
field
of
some
point
of
the
underlying
scheme
X
a
bounded
multiplicative
seminorm
—
cf.,
e.g.,
[Brk1],
§1.1,
§1.2]
and
a
countably
indexed
cofinal
subsystem
S
of
S
such
that
if
Z
log
→
X
log
is
a
member
of
S
,
then,
as
Y
log
→
X
log
ranges
over
the
members
of
S
that
lie
over
Z
log
,
the
discrete
valuations
on
residue
fields
of
points
of
the
underlying
scheme
Z
of
Z
log
determined
by
the
elements
z
(Y
log
)
∈
VCN(Y
log
)
[cf.
the
discussion
preceding
the
present
Corollary
1.15]
converge
in
the
“Berkovich
space
topology”
—
i.e.,
as
bounded
mul-
tiplicative
seminorms
—
to
the
valuation
on
the
residue
field
of
some
point
of
Z
determined
by
v
z
.
log
)
for
the
subset
consisting
of
el-
(iii)
Write
Stab(s)
⊆
VCN(
X
log
)
such
that
the
image
of
s
stabilizes
ements
z
∈
VCN(
X
z
.
Suppose
that
H
is
l-cyclotomically
full
[cf.
(i)].
Then
it
holds
that
Stab(s)
=
∅.
In
particular,
if
z
∈
Stab(s),
and
the
residue
field
of
R
is
countable,
then
the
image
of
s
lies
in
the
decomposition
group
of
any
valuation
v
z
as
in
(ii).
log
→
(iv)
Let
Y
log
→
X
log
be
a
finite
connected
subcovering
of
X
log
X
that
admits
a
stable
model
over
the
normalization
R
Y
of
R
in
Y
;
z
1
,
z
2
∈
Stab(s)
[cf.
(iii)].
Then
one
of
the
following
four
[mutually
exclusive]
conditions
is
satisfied:
z
1
(Y
log
),
z
2
(Y
log
))
≤
(a)
z
1
(Y
log
),
z
2
(Y
log
)
∈
Vert(Y
log
),
and
δ(
2
[cf.
Definition
1.1,
(iii)].
(b)
z
1
(Y
log
),
z
2
(Y
log
)
∈
Edge(Y
log
),
and,
moreover,
V(
z
1
(Y
log
))
∩V(
z
2
(Y
log
))
=
∅.
(c)
z
1
(Y
log
)
∈
Vert(Y
log
),
z
2
(Y
log
)
∈
Edge(Y
log
),
and,
more-
z
1
(Y
log
))∩V(
z
2
(Y
log
))
=
∅
[cf.
Definition
1.10,
over,
V
δ≤1
(
(i)].
(d)
z
1
(Y
log
)
∈
Edge(Y
log
),
z
2
(Y
log
)
∈
Vert(Y
log
),
and,
more-
over,
V(
z
1
(Y
log
))
∩
V
δ≤1
(
z
2
(Y
log
))
=
∅.
(v)
In
the
situation
of
(iv),
suppose,
moreover,
that
the
following
assertion
—
i.e.,
concerning
“resolution
of
nonsingulari-
ties”
[cf.
Remark
1.15.1
below]
—
holds:
(†
RNS
):
Let
Y
log
→
X
log
be
a
finite
connected
sub-
log
→
X
log
that
admits
a
stable
model
covering
of
X
Y
log
over
R
Y
and
y
∈
Y
a
node
of
Y.
Then
there
exists
a
finite
connected
subcovering
Z
log
→
Y
log
of
log
→
Y
log
that
admits
a
stable
model
Z
log
over
R
Z
X
such
that
the
fiber
over
y
of
the
morphism
Z
→
Y
determined
by
Z
log
→
Y
log
is
not
finite.
40
YUICHIRO
HOSHI
AND
SHINICHI
MOCHIZUKI
log
→
Then
every
finite
connected
subcovering
Y
log
→
X
log
of
X
log
X
that
admits
a
stable
model
over
R
Y
satisfies
one
of
the
following
four
[mutually
exclusive]
conditions:
(a
)
z
1
(Y
log
),
z
2
(Y
log
)
∈
Vert(Y
log
),
and
z
1
(Y
log
)
=
z
2
(Y
log
).
(b
)
z
1
(Y
log
),
z
2
(Y
log
)
∈
Edge(Y
log
),
and,
moreover,
V(
z
1
(Y
log
))
log
∩V(
z
2
(Y
))
=
∅.
(c
)
z
1
(Y
log
)
∈
Vert(Y
log
),
z
2
(Y
log
)
∈
Edge(Y
log
),
and,
more-
z
2
(Y
log
)).
over,
z
1
(Y
log
)
∈
V(
(d
)
z
1
(Y
log
)
∈
Edge(Y
log
),
z
2
(Y
log
)
∈
Vert(Y
log
),
and,
more-
z
1
(Y
log
)).
over,
z
2
(Y
log
)
∈
V(
tp
(vi)
Write
Δ
X
log
for
the
Σ-tempered
fundamental
group
of
log
[cf.
[CbTpIII],
Definition
3.1,
(ii)];
Π
tp
for
the
geo-
X
K
X
log
metrically
Σ-tempered
fundamental
group
of
X
log
[i.e.,
the
quotient
of
the
tempered
fundamental
group
of
X
log
by
the
kernel
of
the
natural
surjection
from
the
tempered
fundamen-
log
tal
group
of
X
K
onto
Δ
tp
].
Thus,
we
have
a
natural
exact
X
log
sequence
of
topological
groups
1
−→
Δ
tp
−→
Π
tp
−→
G
K
−→
1.
X
log
X
log
Write
Sect(Π
X
log
/H)
for
the
set
of
Δ
X
log
-conjugacy
classes
of
continuous
sections
of
the
restriction
to
H
⊆
G
K
of
the
natural
surjection
Π
X
log
G
K
and
Sect(Π
tp
/H)
for
the
set
of
Δ
tp
-
X
log
X
log
conjugacy
classes
of
continuous
sections
of
the
restriction
to
G
K
.
Then
the
H
⊆
G
K
of
the
natural
surjection
Π
tp
X
log
natural
map
Sect(Π
tp
/H)
−→
Sect(Π
X
log
/H)
X
log
is
injective.
If,
moreover,
H
is
l-cyclotomically
full
[cf.
(i)],
then
this
map
is
bijective.
Proof.
Assertion
(i)
follows
immediately
from
the
definition
of
the
term
“IPSC-type”
[cf.
[NodNon],
Definition
2.4,
(i)],
together
with
the
well-
known
structure
of
the
maximal
pro-Σ
†
quotient
of
I
K
.
Next,
we
verify
assertion
(ii).
Let
us
first
observe
that
it
follows
immediately
from
our
countability
assumption
on
the
residue
field
of
R
that
the
following
three
assertions
hold:
•
If
Y
log
→
X
log
is
a
member
of
S,
and
z
(Y
log
)
∈
Cusp(Y
log
),
then
the
function
field
of
Y
admits
a
subset
which
is
countable
and
dense,
i.e.,
with
respect
to
the
topology
determined
by
the
discrete
valuation
determined
by
the
element
z
(Y
log
)
∈
VCN(Y
log
).
•
If
Y
log
→
X
log
is
a
member
of
S,
and
z
(Y
log
)
∈
Cusp(Y
log
),
then
the
normalization
R
Y
of
R
in
Y
admits
a
subset
which
COMBINATORIAL
ANABELIAN
TOPICS
IV
41
is
countable
and
dense,
i.e.,
with
respect
to
the
topology
de-
termined
by
the
discrete
valuation
determined
by
the
element
z
(Y
log
)
∈
VCN(Y
log
).
•
There
exists
a
countably
indexed
cofinal
subsystem
of
S
[cf.,
e.g.,
[AbsTpII],
Lemma
2.1].
Thus,
assertion
(ii)
follows
immediately,
by
applying
a
standard
ar-
gument
involving
Cantor
diagonalization,
from
the
well-known
[local]
compactness
of
Berkovich
spaces
[cf.,
e.g.,
[Brk1],
Theorem
1.2.1].
Here,
we
recall
in
passing
that
this
compactness
is,
in
essence,
a
conse-
quence
of
the
compactness
of
a
product
of
copies
of
the
closed
interval
[0,
1]
⊆
R.
This
completes
the
proof
of
assertion
(ii).
We
refer
to
Theorem
A.7
in
Appendix
for
another
approach
to
proving
assertion
(ii).
Assertion
(iii)
follows
immediately
from
the
observation
that,
by
ap-
plying
Theorem
1.13,
(i)
[cf.
also
Remark
1.7.1;
assertion
(i)
of
the
present
Corollary
1.15;
[CmbGC],
Proposition
1.2,
(i)],
together
with
the
well-known
fact
that
a
projective
limit
of
nonempty
finite
sets
is
log
→
X
log
,
nonempty,
to
the
various
finite
connected
subcoverings
of
X
log
fixes
some
one
may
conclude
that
the
action
of
G
K
,
via
s,
on
X
log
)
of
VCN(
X
log
).
[Here,
we
note
that
when
one
element
z
s
∈
VCN(
X
applies
Theorem
1.13,
(i),
to
the
various
finite
connected
subcoverings
log
→
X
log
,
the
conducting
subgroup
“I
G
”
of
Theorem
1.13,
(i),
of
X
must
be
allowed
to
vary
among
suitable
open
subgroups
of
the
origi-
nal
conducting
subgroup
I
G
.]
Assertion
(iv)
follows
immediately
[cf.
also
Remark
1.7.1;
assertion
(i)
of
the
present
Corollary
1.15]
from
Lemma
1.11,
(ii).
Next,
we
verify
assertion
(v).
Let
us
first
observe
that
it
follows
immediately
from
assertion
(iv)
that
if
Y
log
→
X
log
is
a
finite
connected
log
→
X
log
that
admits
a
stable
model
over
R
Y
,
then
subcovering
of
X
log
z
1
(Y
)
and
z
2
(Y
log
)
lie
in
a
connected
sub-semi-graph
Γ
∗
of
Γ
Y
log
such
that
VCN(Γ
∗
)
=
Vert(Γ
∗
)
+
Edge(Γ
∗
)
≤
3
+
2
=
5.
Now
one
verifies
immediately
that
this
uniform
bound
“5”
implies
that
there
exists
a
cofinal
system
S
=
{Y
log
→
X
log
}
consisting
of
finite
log
→
X
log
such
that
Y
log
admits
Galois
subcoverings
Y
log
→
X
log
of
X
a
stable
model
over
R
Y
,
and,
moreover,
Γ
Y
log
admits
a
connected
sub-
semi-graph
Γ
∗
Y
log
such
that
•
z
1
(Y
log
)
and
z
2
(Y
log
)
lie
in
Γ
∗
Y
log
;
•
VCN(Γ
∗
Y
log
)
≤
5;
•
the
semi-graphs
Γ
∗
Y
log
map
isomorphically
to
one
another
as
one
varies
Y
log
→
X
log
.
42
YUICHIRO
HOSHI
AND
SHINICHI
MOCHIZUKI
def
Write
V
∗
(Y
log
)
=
Vert(Γ
∗
Y
log
).
Then
it
follows
immediately
from
asser-
tion
(iv)
that,
to
complete
the
verification
of
assertion
(v),
it
suffices
to
verify
that
the
following
assertion
holds:
Claim
1.15.A:
V
∗
(Y
log
)
≤
1.
Indeed,
suppose
that
V
∗
(Y
log
)
≥
2.
Then
it
follows
immediately
that
there
exists
a
compatible
system
of
nodes
e(Y
log
)
of
Γ
∗
Y
log
[i.e.,
compat-
ible
as
one
varies
Y
log
→
X
log
in
S],
each
of
which
abuts
to
distinct
vertices
v
α
(Y
log
),
v
β
(Y
log
)
of
Γ
∗
Y
log
.
[Thus,
one
may
assume
that
the
vertices
v
α
(−)
(respectively,
v
β
(−))
form
a
compatible
system
of
ver-
tices.]
But
this
implies
that
for
every
Z
log
→
X
log
in
S
that
lies
over
Y
log
→
X
log
in
S,
if
we
write
Y
log
,
Z
log
for
the
respective
stable
mod-
els
of
Y
log
,
Z
log
[so
the
morphism
Z
log
→
Y
log
extends
to
a
morphism
Z
log
→
Y
log
—
cf.,
e.g.,
[ExtFam],
Theorem
C],
then
the
inverse
im-
age
in
Z
log
of
the
node
e(Y
log
)
admits
at
least
one
isolated
point
[i.e.,
e(Z
log
)],
hence
[since
the
covering
Z
log
→
Y
log
is
Galois]
the
entire
in-
verse
image
in
Z
log
of
e(Y
log
)
is
of
dimension
zero.
On
the
other
hand,
this
contradicts
the
assertion
(†
RNS
)
in
the
statement
of
assertion
(v).
This
completes
the
proof
of
assertion
(v).
Finally,
we
verify
assertion
(vi).
The
injectivity
portion
of
assertion
(v)
follows
immediately
from
the
injectivity
portion
of
Theorem
1.13,
(iii)
[cf.
also
Remark
1.7.1;
assertion
(i)
of
the
present
Corollary
1.15],
log
→
X
log
,
applied
to
the
various
finite
connected
subcoverings
of
X
†
where
we
take
the
“Σ”
of
Theorem
1.13
to
be
Σ
[cf.
also
the
fact
that,
in
the
notation
of
Theorem
1.13,
“Π
tp
G
”
is
dense
in
“Π
G
”
in
the
profinite
topology].
Here,
we
note
that
•
when
one
applies
Theorem
1.13,
(iii),
to
the
various
finite
con-
log
→
X
log
,
the
conducting
subgroup
nected
subcoverings
of
X
“I
G
”
of
Theorem
1.13,
(iii),
must
be
allowed
to
vary
among
suitable
open
subgroups
of
the
original
conducting
subgroup
I
G
,
and
that
•
it
follows
immediately
from
the
final
portion
of
Lemma
1.11,
(iv),
that
the
resulting
conjugacy
indeterminacies
that
occur
at
various
subcoverings
are
uniquely
determined
up
to
profinite
centralizers
of
the
sections
that
appear,
hence
converge
in
Δ
tp
X
log
[i.e.,
if
one
passes
to
an
appropriate
subsequence
of
the
system
of
subcoverings
under
consideration].
If
H
is
l-cyclotomically
full,
then
the
surjectivity
of
the
map
/H)
→
Sect(Π
X
log
/H)
Sect(Π
tp
X
log
follows
formally
[cf.
the
proof
of
the
final
portion
of
Theorem
1.13,
(iii)]
from
the
nonemptiness
verified
in
assertion
(iii).
This
completes
the
proof
of
assertion
(vi).
COMBINATORIAL
ANABELIAN
TOPICS
IV
43
Remark
1.15.1.
It
follows
from
[Tama2],
Theorem
0.2,
(v),
that
if
K
is
of
characteristic
zero,
the
residue
field
of
R
is
algebraic
over
F
p
,
and
Σ
=
Primes,
then
the
assertion
(†
RNS
)
in
the
statement
of
Corol-
lary
1.15,
(v),
holds.
Remark
1.15.2.
(i)
Corollary
1.15,
(iii),
(v)
[cf.
also
[SemiAn],
Lemma
5.5],
may
be
regarded
as
a
generalization
of
the
Main
Result
of
[PS].
These
results
are
obtained
in
the
present
paper
[cf.
the
proof
of
Theorem
1.13,
(i)]
by,
in
essence,
combining,
via
a
simi-
lar
argument
to
the
argument
applied
in
the
tempered
case
treated
in
[SemiAn],
Theorems
3.7,
5.4
[cf.
also
the
proof
of
Theorem
1.13,
(ii),
of
the
present
paper],
the
uniqueness
re-
sult
given
in
[NodNon],
Propositions
3.8,
(i);
3.9,
(i),
(ii),
(iii)
[cf.
the
proof
of
Lemma
1.11,
(ii)],
with
the
existence
of
fixed
points
of
actions
of
finite
groups
on
graphs
that
follows
as
a
consequence
of
the
classical
fact
that
[discrete
or
pro-Σ]
free
groups
are
torsion-free
[cf.
Remarks
1.6.2,
1.13.1;
the
proof
of
Lemma
1.6,
(ii)].
One
slight
difference
between
the
profinite
and
tempered
cases
is
that,
whereas,
in
the
tempered
case,
it
follows
from
the
discreteness
of
the
fundamental
groups
of
graphs
that
appear
that
the
actions
of
profinite
groups
on
uni-
versal
coverings
of
such
graphs
necessarily
factor
through
fi-
nite
quotients,
the
corresponding
fact
in
the
profinite
case
is
obtained
as
a
consequence
of
the
fact
that,
under
a
suitable
assumption
on
the
cyclotomic
characters
that
appear,
any
ho-
momorphism
from
a
“positive
slope”
module
to
a
torsion-free
“slope
zero”
module
necessarily
vanishes
[cf.
the
proof
of
Claim
1.13.B
in
Theorem
1.13,
(i)].
That
is
to
say,
in
a
word,
these
results
are
obtained
in
the
present
paper
as
a
consequence
of
abstract
considerations
concerning
abstract
profi-
nite
groups
acting
on
abstract
semi-graphs
that
may,
for
instance,
arise
as
dual
semi-graphs
of
geo-
metric
special
fibers
of
stable
models
of
curves
that
appear
in
scheme
theory,
but,
a
priori,
have
nothing
to
do
with
scheme
theory.
This
a
priori
irrelevance
of
scheme
theory
to
such
abstract
considerations
is
reflected
both
in
the
variety
of
the
results
obtained
in
the
present
§1
as
consequences
of
Theorem
1.13,
as
well
as
in
the
generality
of
Corollary
1.15.
This
approach
contrasts
quite
substantially
with
the
approach
of
[PS],
i.e.,
where
the
main
results
are
derived
as
a
consequence
of
highly
scheme-theoretic
considerations
concerning
stable
curves
over
complete
discrete
valuation
rings,
in
which
the
theory
of
the
44
YUICHIRO
HOSHI
AND
SHINICHI
MOCHIZUKI
Brauer
group
of
the
function
field
of
such
a
curve
plays
a
central
role
[cf.
[PS],
§4].
(ii)
The
essential
equivalence
between
the
issue
of
considering
val-
uations
fixed
by
Galois
actions
and
the
issue
of
considering
vertices
or
edges
of
associated
dual
semi-graphs
fixed
by
Ga-
lois
actions
may
be
seen
in
the
well-known
functorial
homotopy
equivalence
between
the
Berkovich
space
associated
to
a
stable
curve
over
a
complete
discrete
valuation
ring
and
the
associated
dual
graph
[cf.
[Brk2],
Theorems
8.1,
8.2].
Moreover,
the
issue
of
convergence
of
[sub]sequences
of
valuations
fixed
by
Galois
actions
is
an
easy
consequence
of
the
well-known
[local]
com-
pactness
of
Berkovich
spaces
[cf.
the
proof
of
Corollary
1.15,
(ii);
[Brk1],
Theorem
1.2.1],
i.e.,
in
essence,
a
consequence
of
the
well-known
compactness
of
a
product
of
copies
of
the
closed
interval
[0,
1]
⊆
R.
That
is
to
say,
there
is
no
need
to
consider
the
quite
complicated
[and,
at
the
time
of
writing,
not
well
understood!]
structure
of
inductive
limits
of
local
rings,
as
dis-
cussed
in
[PS],
§1.6.
Remark
1.15.3.
Recall
that
in
Corollary
1.15,
(ii),
and
the
final
por-
tion
of
Corollary
1.15,
(iii),
we
assume
that
the
residue
field
of
R
is
countable.
In
fact,
however,
it
is
not
difficult
to
see
that,
in
the
situ-
ation
of
Corollary
1.15,
there
exists
a
complete
discrete
valuation
ring
R
†
that
is
dominated
by
R,
and
whose
residue
field
is
countable
such
that
•
the
smooth
log
curve
X
log
,
•
the
closed
subgroup
H
⊆
G
K
,
and
•
the
section
s
:
H
→
Π
X
log
descend
to
the
field
of
fractions
of
R
†
.
Indeed,
let
us
first
observe
that
since
the
moduli
stack
of
pointed
stable
curves
of
a
given
type
over
Z
is
of
finite
type
over
Z,
there
exists
a
complete
discrete
valuation
ring
R
‡
that
is
dominated
by
R,
and
whose
residue
field
is
countable
such
that
the
smooth
log
curve
X
log
descends
to
the
field
of
fractions
of
R
‡
.
Next,
let
us
observe
that
since
[cf.,
e.g.,
[CanLift],
Proposition
2.3,
(ii)]
the
geometric
fundamental
group
“Δ
X
log
”
associated
to
the
smooth
log
curve
X
log
[i.e.,
over
the
field
of
fractions
of
R]
is
naturally
isomorphic
to
the
geometric
fundamental
group
“Δ
X
log
”
associated
to
the
descended
smooth
log
curve
[i.e.,
over
the
field
of
fractions
of
R
‡
],
it
follows
that
both
of
these
geometric
fundamental
groups
are
topolog-
ically
finitely
generated
[cf.,
e.g.,
[MT],
Proposition
2.2,
(ii)],
and
hence
that
there
exists
a
countably
indexed
open
basis
.
.
.
⊆
U
n+1
⊆
U
n
⊆
.
.
.
⊆
U
2
⊆
U
1
⊆
U
0
=
Δ
X
log
COMBINATORIAL
ANABELIAN
TOPICS
IV
45
of
characteristic
open
subgroups
of
Δ
X
log
.
In
particular,
there
exists
a
complete
discrete
valuation
ring
R
†
that
is
dominated
by
R,
and
whose
residue
field
is
countable
such
that,
for
each
positive
integer
n,
the
finite
collection
of
finite
étale
coverings
[which
are
defined
by
means
of
finitely
many
polynomials,
with
finitely
many
coefficients]
corresponding
to
•
the
finite
quotient
Π
X
log
Q
n
determined
by
the
image
of
the
composite
of
the
conjugation
action
Π
X
log
→
Aut(Δ
X
log
)
and
the
natural
homomorphism
Aut(Δ
X
log
)
→
Aut(Δ
X
log
/U
n
)
and
•
the
subgroup
H
n
⊆
Q
n
obtained
by
forming
the
image
of
the
composite
of
the
section
s
:
H
→
Π
X
log
and
the
natural
surjec-
tive
homomorphism
Π
X
log
Q
n
descends
to
the
field
of
fractions
of
R
†
.
46
YUICHIRO
HOSHI
AND
SHINICHI
MOCHIZUKI
2.
Discrete
combinatorial
anabelian
geometry
In
the
present
§2,
we
introduce
the
notion
of
a
semi-graph
of
temper-
oids
of
HSD-type
[i.e.,
“hyperbolic
surface
decomposition
type”
—
cf.
Definition
2.3,
(iii)]
and
discuss
discrete
versions
of
the
profinite
results
obtained
in
[NodNon],
[CbTpI],
[CbTpII],
[CbTpIII].
A
semi-graph
of
temperoids
of
HSD-type
arises
naturally
from
a
decomposition
[satis-
fying
certain
properties]
of
a
hyperbolic
topological
surface
and
may
be
regarded
as
a
discrete
analogue
of
the
notion
of
a
semi-graph
of
anabelioids
of
PSC-type.
The
main
technical
result
of
the
present
§2
is
Theorem
2.15,
one
immediate
consequence
of
which
is
the
following
[cf.
Corollary
2.19]:
An
isomorphism
of
groups
between
the
discrete
funda-
mental
groups
of
a
pair
of
semi-graphs
of
temperoids
of
HSD-type
arises
from
an
isomorphism
between
the
semi-graphs
of
temperoids
of
HSD-type
if
and
only
if
the
induced
isomorphism
between
profinite
comple-
tions
of
fundamental
groups
arises
from
an
isomor-
phism
between
the
associated
semi-graphs
of
anabe-
lioids
of
pro-Primes
PSC-type.
In
the
present
§2,
let
Σ
be
a
nonempty
set
of
prime
numbers.
Definition
2.1.
(i)
We
shall
refer
to
as
a
semi-graph
of
temperoids
G
a
collection
of
data
as
follows:
•
a
semi-graph
G
[cf.
the
discussion
at
the
beginning
of
[SemiAn],
§1],
•
for
each
vertex
v
of
G,
a
connected
temperoid
G
v
[cf.
[SemiAn],
Definition
3.1,
(ii)],
•
for
each
edge
e
of
G,
a
connected
temperoid
G
e
,
together
with,
for
each
branch
b
∈
e
abutting
to
a
vertex
v,
a
mor-
phism
of
temperoids
b
∗
:
G
e
→
G
v
[cf.
[SemiAn],
Definition
3.1,
(iii)].
We
shall
refer
to
a
semi-graph
of
temperoids
whose
underlying
semi-graph
is
connected
as
a
connected
semi-graph
of
temper-
oids.
Given
two
semi-graphs
of
temperoids,
there
is
an
evident
notion
of
(1-)morphism
[cf.
[SemiAn],
Definition
2.1;
[SemiAn],
Remark
2.4.2]
between
semi-graphs
of
temperoids.
(ii)
Let
T
be
a
connected
temperoid.
We
shall
say
that
a
connected
object
H
of
T
is
Σ-finite
if
there
exists
a
morphism
J
→
H
in
T
such
that
J
is
Galois
[hence
connected
—
cf.
[SemiAn],
Definition
3.1,
(iv)],
and,
moreover,
Aut
T
(J)
is
a
finite
group
whose
order
is
a
Σ-integer
[cf.
the
discussion
entitled
“Num-
bers”
in
§0].
We
shall
say
that
an
object
H
of
T
is
Σ-finite
if
H
is
isomorphic
to
a
disjoint
union
of
finitely
many
connected
COMBINATORIAL
ANABELIAN
TOPICS
IV
47
Σ-finite
objects.
We
shall
say
that
an
object
H
of
T
is
a
finite
object
if
H
is
Primes-finite.
We
shall
write
T
Σ
for
the
connected
anabelioid
[cf.
[GeoAn],
Definition
1.1.1]
ob-
tained
by
forming
the
full
subcategory
of
T
whose
objects
are
the
Σ-finite
objects
of
T
.
Thus,
we
have
a
natural
morphism
of
temperoids
[cf.
Remark
2.1.1
below]
T
−→
T
Σ
.
We
shall
write
def
T
=
T
Primes
[cf.
the
discussion
entitled
“Numbers”
in
§0].
Finally,
we
ob-
serve
that
if
T
=
B
tp
(Π),
where
Π
is
a
tempered
group
[cf.
[SemiAn],
Definition
3.1,
(i)],
and
“B
tp
(−)”
denotes
the
cate-
gory
“B
temp
(−)”
of
the
discussion
at
the
beginning
of
[SemiAn],
§3,
then
T
Σ
may
be
naturally
identified
with
B(Π
Σ
),
i.e.,
the
connected
anabelioid
[cf.
[GeoAn],
Definition
1.1.1;
the
discus-
sion
at
the
beginning
of
[GeoAn],
§1]
determined
by
the
pro-Σ
completion
Π
Σ
of
Π.
(iii)
Let
G
be
a
semi-graph
of
temperoids
[cf.
(i)].
Then,
by
re-
placing
the
connected
temperoids
“G
(−)
”
corresponding
to
the
Σ
vertices
and
edges
“(−)”
by
the
connected
anabelioids
“G
(−)
”
[cf.
(ii)],
we
obtain
a
semi-graph
of
anabelioids,
which
we
de-
note
by
G
Σ
[cf.
[SemiAn],
Definition
2.1].
Thus,
it
follows
immediately
from
the
various
definitions
involved
that
the
various
mor-
Σ
”
of
(ii)
determine
a
natural
morphism
phisms
“G
(−)
→
G
(−)
of
semi-graphs
of
temperoids
[cf.
Remark
2.1.1
below]
G
−→
G
Σ
.
We
shall
write
G
=
G
Primes
.
One
verifies
easily
that
if
G
is
a
connected
semi-graph
of
temperoids
[cf.
(i)],
then
G
Σ
is
a
connected
semi-graph
of
anabelioids.
(iv)
Let
G
be
a
connected
semi-graph
of
temperoids
[cf.
(i)].
Sup-
pose
that
[the
underlying
semi-graph
of]
G
has
at
least
one
vertex.
Then
we
shall
write
def
B(G)
=
B(
G)
def
[cf.
(iii);
the
discussion
following
[SemiAn],
Definition
2.1]
for
the
connected
anabelioid
determined
by
the
connected
semi-
graph
of
anabelioids
G.
48
YUICHIRO
HOSHI
AND
SHINICHI
MOCHIZUKI
(v)
Let
G
be
a
semi-graph
of
temperoids.
Then
we
shall
write
Vert(G),
Cusp(G),
Node(G),
Edge(G),
VCN(G),
V,
C,
N
,
E,
and
δ
for
the
Vert,
Cusp,
Node,
Edge,
VCN,
V,
C,
N
,
E,
and
δ
of
Definition
1.1,
(i),
(ii),
(iii),
applied
to
the
underlying
semi-graph
of
G.
(vi)
Let
G
be
a
connected
semi-graph
of
temperoids
[cf.
(i)].
Sup-
pose
that
[the
underlying
semi-graph
of]
G
has
at
least
one
vertex.
Then
we
shall
write
B
tp
(G)
for
the
category
whose
objects
are
given
by
collections
of
data
{S
v
,
φ
e
}
—
where
v
(respectively,
e)
ranges
over
the
elements
of
Vert(G)
(respectively,
Edge(G))
[cf.
(v)];
for
each
v
∈
Vert(G),
S
v
is
an
object
of
the
temperoid
G
v
corresponding
to
v;
for
each
e
∈
Edge(G),
with
branches
b
1
,
b
2
abutting
to
vertices
v
1
,
v
2
,
∼
respectively,
φ
e
:
((b
1
)
∗
)
∗
S
v
1
→
((b
2
)
∗
)
∗
S
v
2
is
an
isomorphism
in
the
temperoid
G
e
corresponding
to
e
—
and
whose
morphisms
are
given
by
morphisms
[in
the
evident
sense]
between
such
collections
of
data.
In
particular,
the
category
[i.e.,
connected
anabelioid]
B(G)
of
(iv)
may
be
regarded
as
a
full
subcategory
B(G)
⊆
B
tp
(G)
of
B
tp
(G).
One
verifies
immediately
that
any
object
G
of
B
tp
(G)
determines,
in
a
natural
way,
a
semi-graph
of
temper-
oids
G
,
together
with
a
morphism
of
semi-graphs
of
temperoids
G
→
G.
We
shall
refer
to
this
morphism
G
→
G
as
the
cov-
ering
of
G
associated
to
G
.
We
shall
say
that
a
morphism
of
semi-graphs
of
temperoids
is
a
covering
(respectively,
finite
étale
covering)
of
G
if
it
factors
as
the
post-composite
of
an
isomorphism
of
semi-graphs
of
temperoids
with
the
covering
of
G
associated
to
some
object
of
B
tp
(G)
(respectively,
of
B(G)
(⊆
B
tp
(G))).
We
shall
say
that
a
covering
of
G
is
connected
if
the
underlying
semi-graph
of
the
domain
of
the
covering
is
connected.
Remark
2.1.1.
Since
every
profinite
group
is
tempered
[cf.
[SemiAn],
Definition
3.1,
(i);
[SemiAn],
Remark
3.1.1],
it
follows
immediately
that
a
connected
anabelioid
[cf.
[GeoAn],
Definition
1.1.1]
determines,
in
a
natural
way
[i.e.,
by
considering
formal
countable
coproducts,
as
in
the
discussion
entitled
“Categories”
in
[SemiAn],
§0],
a
connected
tem-
peroid
[cf.
[SemiAn],
Definition
3.1,
(ii)].
In
particular,
a
semi-graph
of
anabelioids
[cf.
[SemiAn],
Definition
2.1]
determines,
in
a
natural
way,
a
semi-graph
of
temperoids
[cf.
Definition
2.1,
(i)].
By
abuse
of
COMBINATORIAL
ANABELIAN
TOPICS
IV
49
notation,
we
shall
often
use
the
same
notation
for
the
connected
tem-
peroid
(respectively,
semi-graph
of
temperoids)
naturally
associated
to
a
connected
anabelioid
(respectively,
semi-graph
of
anabelioids).
Definition
2.2.
(i)
Let
T
be
a
topological
space.
Then
we
shall
say
that
a
closed
subspace
of
T
(respectively,
a
closed
subspace
of
T
;
an
open
subspace
of
T
)
is
a
circle
(respectively,
a
closed
disc;
an
open
disc)
on
T
if
it
is
homeomorphic
to
the
set
{
(s,
t)
∈
R
2
|
s
2
+
t
2
=
1
}
(respectively,
{
(s,
t)
∈
R
2
|
s
2
+
t
2
≤
1
};
{
(s,
t)
∈
R
2
|
s
2
+
t
2
<
1
})
equipped
with
the
topology
induced
by
the
topology
of
R
2
.
If
D
⊆
T
is
a
closed
disc
on
T
,
then
we
shall
write
∂D
⊆
D
for
the
circle
on
T
determined
by
the
boundary
of
D
regarded
as
a
two-dimensional
topological
manifold
with
boundary
[i.e.,
the
closed
subspace
of
D
corresponding
to
the
closed
subspace
{
(s,
t)
∈
R
2
|
s
2
+
t
2
=
1
}
⊆
{
(s,
t)
∈
R
2
|
s
2
+
def
t
2
≤
1
}]
and
D
◦
=
D
\
∂D
⊆
D
for
the
open
disc
on
T
obtained
by
forming
the
complement
of
∂D
in
D.
(ii)
Let
(g,
r)
be
a
pair
of
nonnegative
integers.
Then
we
shall
say
that
a
pair
X
=
(X,
{D
i
}
ri=1
)
consisting
of
a
connected
orientable
compact
topological
surface
X
of
genus
g
and
a
col-
lection
of
r
disjoint
closed
discs
D
i
⊆
X
of
X
[cf.
(i)]
is
of
HS-type
[where
the
“HS”
stands
for
“hyperbolic
surface”]
if
2g
−
2
+
r
>
0.
(iii)
Let
X
=
(X,
{D
i
}
ri=1
)
be
a
pair
of
HS-type
[cf.
(ii)].
Then
we
shall
write
r
◦
def
U
X
=
X\
D
i
i=1
[cf.
(i)]
and
refer
to
U
X
as
the
interior
of
X.
We
shall
refer
to
a
circle
on
U
X
determined
by
some
∂D
i
⊆
U
X
[cf.
(i)]
as
a
cusp
of
U
X
,
or
alternatively,
X.
Write
∂U
X
⊆
U
X
for
the
union
of
the
cusps
of
U
X
;
I
X
for
the
group
of
homeomorphisms
∼
φ
:
X
→
X
such
that
φ
restricts
to
the
identity
on
U
X
.
Suppose
that
Y
=
(Y
,
{E
i
}
sj=1
)
is
also
a
pair
of
HS-type.
Then
we
define
∼
an
isomorphism
X
→
Y
of
pairs
of
HS-type
to
be
an
I
X
-orbit
∼
of
homeomorphisms
X
→
Y
such
that
each
homeomorphism
ψ
∼
that
belongs
to
the
I
X
-orbit
induces
a
homeomorphism
U
X
→
U
Y
.
(iv)
Let
X
=
(X,
{D
i
}
ri=1
)
be
a
pair
of
HS-type
[cf.
(ii)]
and
{Y
j
}
j∈J
a
finite
collection
of
pairs
of
HS-type.
For
each
j
∈
J,
let
ι
j
:
U
Y
j
→
U
X
[cf.
(iii)]
be
a
local
immersion
[i.e.,
a
map
that
restricts
to
a
homeomorphism
between
some
open
neigh-
borhood
of
each
point
of
the
domain
and
the
image
of
the
50
YUICHIRO
HOSHI
AND
SHINICHI
MOCHIZUKI
open
neighborhood,
equipped
with
the
induced
topology,
in
the
codomain]
of
topological
spaces.
Then
we
shall
say
that
a
pair
({Y
j
}
j∈J
,
{ι
j
}
j∈J
)
is
an
HS-decomposition
of
X
if
the
following
conditions
are
satisfied:
(1)
U
X
=
j∈J
ι
j
(U
Y
j
).
(2)
For
any
j
∈
J,
the
complement
of
the
diagonal
in
U
Y
j
×
U
X
U
Y
j
is
a
disjoint
union
of
circles,
each
of
which
maps
home-
omorphically,
via
the
two
projections
to
U
Y
j
,
to
two
dis-
tinct
cusps
of
U
Y
j
[cf.
(iii)].
[Thus,
by
“Brouwer
invariance
of
domain”,
it
follows
that
ι
j
restricts
to
an
open
immer-
sion
on
the
complement
of
the
cusps
of
U
Y
j
.]
(3)
For
any
j,
j
∈
J
such
that
j
=
j
,
every
connected
com-
ponent
of
U
Y
j
×
U
X
U
Y
j
projects
homeomorphically
onto
cusps
of
U
Y
j
and
U
Y
j
.
(4)
For
any
[i.e.,
possibly
equal]
j,
j
∈
J,
we
shall
refer
to
a
circle
of
U
Y
j
×
U
X
U
Y
j
that
forms
a
connected
component
of
U
Y
j
×
U
X
U
Y
j
as
a
pre-node
[of
the
HS-decomposition
({Y
j
}
j∈J
,
{ι
j
}
j∈J
)]
and
to
the
cusps
of
U
Y
j
,
U
Y
j
that
arise
as
the
images
of
such
a
pre-node
via
the
projections
to
U
Y
j
,
U
Y
j
as
the
branch
cusps
of
the
pre-node.
Then
we
suppose
further
that
every
pre-node
maps
injectively
into
U
X
,
and
that
the
image
in
U
X
of
the
pre-node
has
empty
intersec-
tion
with
∂U
X
,
as
well
as
with
the
image
via
ι
j
,
for
j
∈
J,
of
any
cusp
of
U
Y
j
which
is
not
a
branch
cusp
of
the
pre-
node.
We
shall
refer
to
the
image
in
U
X
of
a
pre-node
as
a
node
[of
the
HS-decomposition
({Y
j
}
j∈J
,
{ι
j
}
j∈J
)].
Thus,
[one
verifies
easily
that]
every
node
arises
from
a
unique
pre-node.
We
shall
refer
to
the
branch
cusps
of
the
pre-
node
that
gives
rise
to
a
node
as
the
branch
cusps
of
the
node.
[Thus,
by
“Brouwer
invariance
of
domain”,
it
fol-
lows
that,
for
any
pre-node
of
U
Y
j
×
U
X
U
Y
j
,
the
maps
ι
j
,
ι
j
determine
a
homeomorphism
of
the
topological
space
ob-
tained
by
gluing,
along
the
associated
node,
suitable
open
neighborhoods
of
the
branch
cusps
of
U
Y
j
,
U
Y
j
onto
the
topological
space
constituted
by
a
suitable
open
neighbor-
hood
of
the
associated
node
in
U
X
.]
(5)
For
any
j
∈
J,
every
cusp
of
U
Y
j
maps
homeomorphically
onto
either
a
cusp
of
U
X
or
a
node
of
({Y
j
}
j∈J
,
{ι
j
}
j∈J
)
[cf.
(4)].
Moreover,
every
cusp
of
U
X
arises
in
this
way
from
a
cusp
of
U
Y
j
for
some
[necessarily
uniquely
determined]
j
∈
J.
[Thus,
by
“Brouwer
invariance
of
domain”
—
together
with
a
suitable
gluing
argument
as
in
(4)
—
it
follows
that
every
cusp
of
U
X
admits
an
open
neighborhood
that
COMBINATORIAL
ANABELIAN
TOPICS
IV
51
arises,
for
some
j
∈
J,
as
the
homeomorphic
image,
via
ι
j
,
of
a
suitable
open
neighborhood
of
a
cusp
of
U
Y
j
.]
If
({Y
j
},
{ι
j
})
is
an
HS-decomposition
of
X,
then
we
shall
re-
fer
to
the
triple
(X,
{Y
j
},
{ι
j
})
as
a
collection
of
HSD-data
[where
the
“HSD”
stands
for
“hyperbolic
surface
decomposi-
tion”].
If
X
=
(X,
{Y
j
},
{ι
j
})
is
a
collection
of
HSD-data,
then
we
shall
refer
to
the
topological
space
U
X
(respectively,
[the
closed
subspace
of
U
X
corresponding
to]
an
element
of
the
[fi-
nite]
set
{Y
j
};
a
cusp
of
U
X
;
a
node
of
({Y
j
},
{ι
j
})
[cf.
(4)])
as
the
underlying
surface
(respectively,
a
vertex;
a
cusp;
a
node)
of
X.
Also,
we
shall
refer
to
a
cusp
or
node
of
X
as
an
edge
of
X.
Definition
2.3.
Let
X
=
(X,
{Y
j
},
{ι
j
})
be
a
collection
of
HSD-data
[cf.
Definition
2.2,
(iv)].
(i)
We
shall
refer
to
the
semi-graph
G
X
defined
as
follows
as
the
dual
semi-graph
of
X:
We
take
the
set
of
vertices
(respectively,
open
edges;
closed
edges)
of
G
X
is
the
[finite]
set
of
vertices
(respectively,
cusps;
nodes)
of
X
[cf.
Definition
2.2,
(iv)].
For
a
vertex
v
and
an
edge
e
of
X,
we
take
the
set
of
branches
of
e
that
abut
to
v
to
be
the
set
of
natural
inclusions
[i.e.,
that
arise
from
X
—
cf.
Definition
2.2,
(iv)]
from
the
edge
of
X
corresponding
to
e
into
the
topological
space
U
Y
j
associated
to
the
Y
j
corresponding
to
the
vertex
v.
(ii)
We
shall
refer
to
the
connected
semi-graph
G
X
of
temperoids
[cf.
Definition
2.1,
(i)]
defined
as
follows
as
the
semi-graph
of
temperoids
associated
to
X:
We
take
the
underly-
ing
semi-graph
of
G
X
to
be
G
X
[cf.
(i)].
For
each
vertex
v
of
G
X
,
we
take
the
connected
temperoid
of
G
X
corresponding
to
v
to
be
the
connected
temperoid
determined
by
the
category
of
topo-
logical
coverings
with
countably
many
connected
components
of
the
topological
space
U
Y
j
[cf.
Definition
2.2,
(iii)]
associated
to
the
Y
j
corresponding
to
the
vertex
v.
For
each
edge
e
of
G
X
,
we
take
the
connected
temperoid
of
G
X
corresponding
to
e
to
be
the
connected
temperoid
determined
by
the
category
of
topological
coverings
with
countably
many
connected
compo-
nents
of
the
circle
[cf.
Definition
2.2,
(i)]
on
U
X
corresponding
to
the
edge
e.
For
each
branch
b
of
G
X
,
we
take
the
morphism
of
temperoids
corresponding
to
b
to
be
the
morphism
obtained
by
pulling
back
topological
coverings
of
the
topological
spaces
under
consideration.
52
YUICHIRO
HOSHI
AND
SHINICHI
MOCHIZUKI
(iii)
We
shall
say
that
a
semi-graph
of
temperoids
is
of
HSD-type
if
it
is
isomorphic
to
the
semi-graph
of
temperoids
associated
to
some
collection
of
HSD-data
[cf.
(ii)].
Example
2.4
(Semi-graphs
of
temperoids
of
HSD-type
asso-
ciated
to
stable
log
curves).
Let
(g,
r)
be
a
pair
of
nonnegative
def
integers
such
that
2g
−
2
+
r
>
0.
Write
S
=
Spec(C).
In
the
fol-
lowing,
we
shall
apply
the
notation
and
terminology
of
the
discussion
entitled
“Curves”
in
[CbTpI],
§0.
(i)
Let
S
→
(M
g,r
)
C
be
a
C-valued
point
of
(M
g,r
)
C
.
Write
S
log
for
the
fs
log
scheme
obtained
by
equipping
S
with
the
log
log
structure
induced
by
the
log
structure
of
(M
g,r
)
C
;
X
log
→
S
log
for
the
stable
log
curve
over
S
log
corresponding
to
the
resulting
log
strict
(1-)morphism
S
log
→
(M
g,r
)
C
;
d
for
the
rank
of
the
group-characteristic
of
S
log
[cf.
[MT],
Definition
5.1,
(i)],
i.e.,
log
log
→
S
an
for
the
morphism
of
fs
the
number
of
nodes
of
X
log
;
X
an
log
analytic
spaces
determined
by
the
morphism
X
log
→
S
log
;
X
an
→
S
an
for
the
underlying
morphism
of
analytic
spaces
of
log
log
log
log
→
S
an
;
X
an
(C),
S
an
(C)
for
the
respective
topological
X
an
log
spaces
“X
”
defined
in
[KN],
(1.2),
in
the
case
where
we
take
log
log
the
“X”
of
[KN],
(1.2),
to
be
X
an
,
S
an
,
i.e.,
for
T
∈
{X,
S},
def
log
T
an
(C)
=
{
(t,
h)
|
t
∈
T
an
,
h
∈
Hom
gp
(M
T
gp
an
,t
,
S
1
)
such
that
h(f
)
=
f
(t)/|f
(t)|
for
every
f
∈
O
T
×
an
,t
⊆
M
T
gp
an
,t
}
def
—
where
we
write
S
1
=
{
u
∈
C
|
|u|
=
1
}
and
M
T
an
for
the
sheaf
of
monoids
on
T
an
that
defines
the
log
structure
of
log
.
Then,
by
considering
the
functoriality
discussed
in
[KN],
T
an
log
log
(1.2.5),
and
the
respective
maps
X
an
(C)
→
X
an
,
S
an
(C)
→
S
an
induced
by
the
first
projections,
we
obtain
a
commutative
diagram
of
topological
spaces
and
continuous
maps
log
X
an
(C)
−−−→
X
an
⏐
⏐
⏐
⏐
log
(C)
−−−→
S
an
.
S
an
Now
one
verifies
immediately
from
the
various
definitions
in-
log
volved
that
S
an
(C)
is
homeomorphic
to
a
product
(S
1
)
×d
of
d
copies
of
S
1
;
moreover,
it
follows
from
[NO],
Theorem
5.1,
that
the
left-hand
vertical
arrow
of
the
above
diagram
is
a
log
(C).
Thus,
since
[one
ver-
topological
fiber
bundle.
Let
s
∈
S
an
1
×d
ifies
easily
that]
(S
)
is
an
Eilenberg-Maclane
space
[i.e.,
its
COMBINATORIAL
ANABELIAN
TOPICS
IV
53
universal
covering
space
is
contractible],
the
left-hand
vertical
arrow
of
the
above
diagram
determines
an
exact
sequence
log
log
log
1
−→
π
1
(X
an
(C)|
s
)
−→
π
1
(X
an
(C))
−→
π
1
(S
an
(C))
(
∼
=
Z
⊕d
)
−→
1
log
—
where
we
write
X
an
(C)|
s
for
the
fiber
of
the
left-hand
ver-
log
log
tical
arrow
X
an
(C)
→
S
an
(C)
of
the
above
diagram
at
s
—
which
thus
determines
an
outer
action
log
log
π
1
(S
an
(C))
(
∼
(C)|
s
)).
=
Z
⊕d
)
−→
Out(π
1
(X
an
Write
N
⊆
X
an
for
the
finite
subset
consisting
of
the
nodes
of
log
,
C
⊆
X
an
for
the
finite
subset
consisting
of
the
cusps
of
X
an
def
log
X
an
,
U
=
X
an
\
(N
∪
C)
⊆
X
an
,
and
π
0
(U
)
for
the
finite
set
of
connected
components
of
U
.
For
each
node
x
∈
N
(respectively,
cusp
y
∈
C;
connected
component
F
∈
π
0
(U
)
of
U
),
write
C
x
log
(respectively,
C
y
;
Y
F
)
⊆
X
an
(C)|
s
for
the
closure
of
the
inverse
image
of
{x}
(respectively,
{y};
F
)
⊆
X
an
via
the
composite
pr
log
log
X
an
(C)|
s
→
1
X
an
(C)
→
X
an
—
where
the
second
arrow
is
the
upper
horizontal
arrow
of
the
above
diagram.
Then
one
verifies
immediately
from
the
various
definitions
involved
that
there
exists
a
uniquely
determined,
up
to
unique
isomorphism
[in
the
evident
sense],
collection
of
data
as
follows:
•
a
pair
of
HS-type
Z
=
(Z,
{D
i
}
ri=1
)
of
type
(g,
r)
[cf.
Def-
inition
2.2,
(ii)];
∼
log
log
•
a
homeomorphism
φ
:
X
an
(C)|
s
→
U
Z
of
X
an
(C)|
s
with
the
interior
U
Z
of
Z
[cf.
Definition
2.2,
(iii)]
such
that
log
φ
restricts
to
a
homeomorphism
of
y∈C
C
y
⊆
X
an
(C)|
s
r
with
∂U
Z
=
i=1
∂D
i
⊆
U
Z
[cf.
Definition
2.2,
(iii)].
Moreover,
there
exists
a
uniquely
determined,
up
to
unique
isomorphism
[in
the
evident
sense],
HS-decomposition
of
Z
[cf.
Definition
2.2,
(iv)]
such
that
the
set
of
vertices
(respectively,
nodes;
cusps)
[cf.
Definition
2.2,
(iv)]
of
the
resulting
collec-
tion
of
HSD-data
[cf.
Definition
2.2,
(iv)]
is
{φ(Y
F
)}
F
∈π
0
(U
)
(respectively,
{φ(C
x
)}
x∈N
;
{φ(C
y
)}
y∈C
).
We
shall
write
G
X
log
for
the
semi-graph
of
temperoids
of
HSD-type
associated
to
this
collection
of
HSD-data
[cf.
Definition
2.3,
(ii)]
and
refer
to
G
X
log
as
the
semi-graph
of
temperoids
of
HSD-type
associated
to
X
log
.
Then
one
verifies
immediately
from
the
functoriality
discussed
in
[KN],
(1.2.5),
applied
to
the
vertices,
nodes,
and
cusps
of
the
data
under
consideration,
that
the
locally
trivial
log
log
fibration
X
an
(C)
→
S
an
(C)
determines
an
action
log
π
1
(S
an
(C))
(
∼
=
Z
⊕d
)
−→
Aut(G
X
log
),
54
YUICHIRO
HOSHI
AND
SHINICHI
MOCHIZUKI
which
is
compatible,
in
the
evident
sense,
with
the
outer
action
log
log
π
1
(S
an
(C))
−→
Out(π
1
(X
an
(C)|
s
))
discussed
above.
(ii)
Let
S
log
be
the
fs
log
scheme
obtained
by
equipping
S
with
the
log
structure
given
by
the
fs
chart
N
1
→
0
∈
C
and
X
log
→
S
log
a
stable
log
curve
of
type
(g,
r)
over
S
log
[cf.
[CmbGC],
Example
2.5,
in
the
case
where
k
=
C].
Then
one
verifies
easily
log
that
the
classifying
(1-)morphism
S
log
→
(M
g,r
)
C
of
X
log
→
log
S
log
factors
as
a
composite
S
log
→
T
log
→
(M
g,r
)
C
—
where
the
first
arrow
is
a
morphism
that
induces
an
isomorphism
between
the
underlying
schemes,
and
the
second
arrow
is
strict
—
and,
moreover,
if
we
write
Y
log
→
T
log
for
the
stable
log
log
curve
determined
by
the
strict
(1-)morphism
T
log
→
(M
g,r
)
C
,
then
we
have
a
natural
isomorphism
over
S
log
∼
X
log
−→
Y
log
×
T
log
S
log
.
We
shall
write
def
G
X
log
=
G
Y
log
[cf.
(i)]
and
refer
to
G
X
log
as
the
semi-graph
of
temperoids
of
HSD-type
associated
to
X
log
.
Then,
by
pulling
back
the
ac-
tion
of
the
second
to
last
display
of
(i)
via
the
homomor-
log
log
phism
π
1
(S
an
(C))
→
π
1
(T
an
(C))
induced
by
the
morphism
log
log
S
→
T
,
we
obtain
an
action
π
1
(S
log
(C))
(
∼
=
Z)
−→
Aut(G
X
log
),
an
together
with
a
compatible
outer
action
log
log
π
1
(S
an
(C))
−→
Out(π
1
(X
an
(C)|
s
)).
Remark
2.4.1.
One
verifies
easily
that
the
discussion
of
Example
2.4,
(ii),
generalizes
immediately
to
the
case
of
arbitrary
fs
log
schemes
S
log
with
underlying
scheme
S
=
Spec(C).
Proposition
2.5
(Fundamental
groups
of
semi-graphs
of
tem-
peroids
of
HSD-type).
Let
G
be
a
semi-graph
of
temperoids
of
HSD-
type
associated
[cf.
Definition
2.3,
(ii),
(iii)]
to
a
collection
of
HSD-data
X
[cf.
Definition
2.2,
(iv)].
Write
U
X
for
the
underlying
surface
of
X
[cf.
Definition
2.2,
(iv)]
and
B
tp
(U
X
)
for
the
connected
temperoid
[cf.
[SemiAn],
Definition
3.1,
(ii)]
deter-
mined
by
the
category
of
topological
coverings
with
countably
many
con-
nected
components
of
the
topological
space
U
X
.
Then
the
following
hold:
COMBINATORIAL
ANABELIAN
TOPICS
IV
55
(i)
We
have
a
natural
equivalence
of
categories
∼
B
tp
(U
X
)
−→
B
tp
(G)
[cf.
Definition
2.1,
(vi)].
In
particular,
B
tp
(G)
is
a
connected
temperoid.
Write
Π
G
for
the
tempered
fundamental
group
[which
is
well-defined,
up
to
inner
automorphism]
of
the
connected
temperoid
B
tp
(G)
[cf.
[SemiAn],
Remark
3.2.1;
the
discussion
of
“Galois-countable
temperoids”
in
[IUTeichI],
Remark
2.5.3,
(i)].
[Thus,
the
tem-
pered
group
Π
G
admits
a
natural
outer
isomorphism
with
the
topological
fundamental
group,
equipped
with
the
discrete
topol-
ogy,
of
the
topological
space
U
X
.]
We
shall
refer
to
this
tem-
pered
group
Π
G
as
the
fundamental
group
of
G.
(ii)
Every
connected
finite
étale
covering
H
→
G
[cf.
Definition
2.1,
(vi)]
admits
a
natural
structure
of
semi-graph
of
temper-
oids
of
HSD-type.
(iii)
The
connected
semi-graph
of
anabelioids
G
Σ
[cf.
Definition
2.1,
(iii)]
is
of
pro-Σ
PSC-type
[cf.
[CmbGC],
Definition
1.1,
(i)].
Write
Π
G
Σ
for
the
[pro-Σ]
fundamental
group
of
G
Σ
.
Then
the
natural
morphism
G
→
G
Σ
of
semi-graphs
of
temperoids
of
Definition
2.1,
(iii),
induces
a
natural
outer
injection
Π
G
→
Π
G
Σ
[cf.
(i)].
Moreover,
this
natural
outer
injection
determines
an
outer
isomorphism
∼
Π
G
Σ
−→
Π
G
Σ
—
where
we
write
Π
Σ
G
for
the
pro-Σ
completion
of
Π
G
.
(iv)
Let
z
∈
VCN(G)
[cf.
Definition
2.1,
(v)].
Write
Π
G
z
for
the
tempered
fundamental
group
[cf.
[SemiAn],
Remark
3.2.1]
of
the
connected
temperoid
G
z
of
G
corresponding
to
z.
Then
the
natural
outer
homomorphism
Π
G
z
−→
Π
G
is
a
Σ-compatible
injection
[cf.
the
discussion
entitled
“Groups”
in
§0].
(v)
In
the
notation
of
(iii)
and
(iv),
the
closure
of
the
image
of
the
composite
Π
G
z
→
Π
G
→
Π
G
Σ
of
the
outer
injections
of
(iii)
and
(iv)
is
a
VCN-subgroup
of
Π
G
Σ
[cf.
(iii);
[CbTpI],
Definition
2.1,
(i)]
associated
to
z
∈
VCN(G)
=
VCN(G
Σ
).
56
YUICHIRO
HOSHI
AND
SHINICHI
MOCHIZUKI
Proof.
A
natural
equivalence
of
categories
as
in
assertion
(i)
may
be
obtained
by
observing
that,
after
sorting
through
the
various
defini-
tions
involved,
an
object
of
B
tp
(U
X
)
[i.e.,
a
topological
covering
of
U
X
]
amounts
to
the
same
data
as
an
object
of
B
tp
(G).
Assertion
(ii)
follows
immediately
from
the
various
definitions
involved.
Next,
we
verify
assertion
(iii).
The
assertion
that
G
Σ
is
of
pro-Σ
PSC-
type,
as
well
as
the
assertion
that
the
morphism
G
→
G
Σ
determines
an
∼
outer
isomorphism
Π
Σ
G
→
Π
G
Σ
,
follows
immediately
from
the
various
definitions
involved.
Thus,
the
assertion
that
the
morphism
G
→
G
Σ
determines
an
outer
injection
Π
G
→
Π
G
Σ
follows
from
the
well-known
fact
that
the
discrete
group
Π
G
injects
into
its
pro-l
completion
for
any
l
∈
Primes
[cf.,
e.g.,
[RZ],
Proposition
3.3.15;
[Prs],
Theorem
1.7].
Next,
we
verify
the
injectivity
portion
of
assertion
(iv).
Let
us
first
observe
that
it
follows
immediately
from
the
various
definitions
involved
that
the
composite
Π
G
z
→
Π
G
→
Π
G
[cf.
Definition
2.1,
(iii)]
of
the
outer
homomorphism
under
consideration
and
the
outer
injection
of
assertion
(iii)
[in
the
case
where
Σ
=
Primes]
factors
as
the
composite
Π
G
z
→
Π
G
z
→
Π
G
of
the
outer
homomorphism
Π
G
z
→
Π
G
z
induced
by
the
morphism
G
z
→
G
z
of
Definition
2.1,
(ii),
and
the
natural
outer
inclusion
Π
G
z
→
Π
G
[cf.
[SemiAn],
Proposition
2.5,
(i)].
Thus,
to
complete
the
verification
of
the
injectivity
portion
of
assertion
(iv),
it
suffices
to
verify
that
the
outer
homomorphism
Π
G
z
→
Π
G
z
is
injective.
On
the
other
hand,
this
follows
from
the
well-known
fact
that
Π
G
z
injects
into
its
pro-
l
completion
for
any
l
∈
Primes
[cf.,
e.g.,
[RZ],
Proposition
3.3.15;
[Prs],
Theorem
1.7].
This
completes
the
proof
of
the
injectivity
portion
of
assertion
(iv).
Assertion
(v)
follows
immediately
from
the
various
definitions
involved.
Finally,
it
follows
immediately
from
assertions
(iii)
and
(v),
together
with
the
evident
pro-Σ
analogue
of
[SemiAn],
Proposition
2.5,
(i),
that
the
natural
outer
injection
of
assertion
(iv)
is
Σ-compatible.
This
completes
the
proof
of
assertion
(iv),
hence
also
of
Proposition
2.5.
Remark
2.5.1.
In
the
notation
of
Proposition
2.5,
as
is
discussed
in
Proposition
2.5,
(i),
the
fundamental
group
Π
G
of
the
semi-graph
of
temperoids
of
HSD-type
G
is
naturally
isomorphic,
up
to
inner
auto-
morphism,
to
the
topological
fundamental
group,
equipped
with
the
discrete
topology,
of
the
compact
orientable
hyperbolic
topological
sur-
face
with
compact
boundary
U
X
.
In
particular,
Π
G
is
finitely
generated,
torsion-free,
and
center-free
and
injects
into
its
pro-l
completion
for
COMBINATORIAL
ANABELIAN
TOPICS
IV
57
any
l
∈
Primes
[cf.
Proposition
2.5,
(iii)].
Moreover,
it
holds
that
Cusp(G)
=
∅
[cf.
Definition
2.1,
(v)]
if
and
only
if
Π
G
is
free.
Remark
2.5.2.
In
the
situation
of
Example
2.4,
(ii),
write
G
X
log
for
the
Σ
semi-graph
of
temperoids
of
HSD-type
associated
to
X
log
;
G
X
log
for
the
semi-graph
of
anabelioids
of
pro-Σ
PSC-type
of
Proposition
2.5,
(iii),
in
the
case
where
we
take
the
“G”
of
Proposition
2.5,
(iii),
to
be
G
X
log
;
PSC-Σ
for
the
semi-graph
of
anabelioids
of
pro-Σ
PSC-type
associated
G
X
log
log
to
X
[cf.
[CmbGC],
Example
2.5].
Then
it
follows
from
Proposi-
∼
tion
2.5,
(iii),
that
we
have
a
natural
outer
isomorphism
Π
Σ
G
X
log
→
Π
G
Σ
log
.
On
the
other
hand,
by
associating
finite
étale
coverings
of
X
log
(C)
to
log
étale
coverings
of
Kummer
type
of
X
log
[cf.
[KN],
Lemma
X
an
log
2.2]
and
then
restricting
such
finite
étale
coverings
to
X
an
(C)|
s
[cf.
Ex-
Σ
.
ample
2.4,
(i)],
we
obtain
an
outer
homomorphism
Π
G
log
→
Π
G
PSC-Σ
X
X
log
Then
one
verifies
immediately
from
the
various
definitions
involved
that
the
composite
of
the
two
outer
homomorphisms
∼
Π
G
Σ
log
←−
Π
Σ
G
log
−→
Π
G
PSC-Σ
log
X
X
X
is
a
graphic
outer
isomorphism
[cf.
[CmbGC],
Definition
1.4,
(i)],
i.e.,
arises
from
a
uniquely
determined
isomorphism
of
semi-graphs
of
an-
abelioids
∼
Σ
PSC-Σ
G
X
.
log
−→
G
X
log
Finally,
one
verifies
easily
that
the
above
discussion
generalizes
im-
mediately
to
the
case
of
arbitrary
fs
log
schemes
S
log
with
underlying
scheme
S
=
Spec(C)
[cf.
Remark
2.4.1].
Definition
2.6.
Let
G
be
a
semi-graph
of
temperoids
of
HSD-type.
Write
Π
G
for
the
fundamental
group
of
G.
(i)
Let
z
∈
VCN(G)
[cf.
Definition
2.1,
(v)].
Then
we
shall
refer
to
a
closed
subgroup
of
Π
G
that
belongs
to
the
Π
G
-conjugacy
class
of
closed
subgroups
determined
by
the
image
of
the
outer
injection
of
the
display
of
Proposition
2.5,
(iv),
as
a
VCN-
subgroup
of
Π
G
associated
to
z
∈
VCN(G).
If,
moreover,
z
∈
Vert(G)
(respectively,
∈
Cusp(G);
∈
Node(G);
∈
Edge(G))
[cf.
Definition
2.1,
(v)],
then
we
shall
refer
to
a
VCN-subgroup
of
Π
G
associated
to
z
as
a
verticial
(respectively,
a
cuspidal;
a
nodal;
an
edge-like)
subgroup
of
Π
G
associated
to
z.
(ii)
Write
G
→
G
for
the
universal
covering
of
G
correspond-
[cf.
Definition
2.1,
(v)].
Then
ing
to
Π
G
.
Let
z
∈
VCN(
G)
we
shall
refer
to
the
VCN-subgroup
Π
z
⊆
Π
G
[cf.
(i)]
deter-
as
the
VCN-subgroup
of
Π
G
associ-
mined
by
z
∈
VCN(
G)
If,
moreover,
z
∈
Vert(
G)
(respectively,
ated
to
z
∈
VCN(
G).
58
YUICHIRO
HOSHI
AND
SHINICHI
MOCHIZUKI
∈
Node(
G);
∈
Edge(
G))
[cf.
Definition
2.1,
(v)],
∈
Cusp(
G);
then
we
shall
refer
to
the
VCN-subgroup
of
Π
G
associated
to
z
as
the
verticial
(respectively,
cuspidal;
nodal;
edge-like)
sub-
group
of
Π
G
associated
to
z
.
(iii)
Let
(g,
r)
be
a
pair
of
nonnegative
integers
such
that
2g−2+r
>
0
and
v
∈
Vert(G).
Then
we
shall
say
that
v
is
of
type
(g,
r)
if
the
“(g,
r)”
appearing
in
Definition
2.2,
(ii),
for
the
pair
of
HS-type
corresponding
to
v
coincides
with
(g,
r).
Thus,
one
verifies
easily
that
v
is
of
type
(g,
r)
if
and
only
if
the
number
of
the
branches
of
edges
of
G
that
abut
to
v
is
equal
to
r,
and,
moreover,
rank
Z
(Π
ab
v
)
=
2g
+
max{0,
r
−
1}
—
where
we
use
the
notation
Π
v
to
denote
a
verticial
subgroup
associated
to
v.
Remark
2.6.1.
In
the
notation
of
Definition
2.6,
it
follows
from
Propo-
sition
2.5,
(iv),
that
every
verticial
subgroup
of
Π
G
is
naturally
isomor-
phic,
up
to
inner
automorphism,
to
the
topological
fundamental
group,
equipped
with
the
discrete
topology,
of
a
compact
orientable
hyperbolic
topological
surface
with
compact
boundary.
In
particular,
every
verti-
cial
subgroup
of
Π
G
is
finitely
generated,
torsion-free,
and
center-free
and
injects
into
its
pro-l
completion
for
any
l
∈
Primes
[cf.
Proposi-
tion
2.5,
(iii)].
Moreover,
it
follows
from
Proposition
2.5,
(iv),
that
every
edge-like
subgroup
of
Π
G
is
naturally
isomorphic,
up
to
inner
au-
tomorphism,
to
the
topological
fundamental
group,
equipped
with
the
discrete
topology,
of
a
unit
circle
[hence
isomorphic
to
Z].
Definition
2.7.
Let
G
and
H
be
semi-graphs
of
temperoids
of
HSD-
type.
Write
Π
G
,
Π
H
for
the
fundamental
groups
of
G,
H,
respectively.
∼
(i)
We
shall
say
that
an
isomorphism
of
groups
Π
G
→
Π
H
is
group-
theoretically
verticial
(respectively,
group-theoretically
cuspi-
dal;
group-theoretically
nodal)
if
the
isomorphism
induces
a
bijection
between
the
set
of
the
verticial
(respectively,
cusp-
idal;
nodal)
subgroups
[cf.
Definition
2.6,
(i)]
of
Π
G
and
the
set
of
the
verticial
(respectively,
cuspidal;
nodal)
subgroups
of
Π
H
.
∼
We
shall
say
that
an
outer
isomorphism
Π
G
→
Π
H
is
group-
theoretically
verticial
(respectively,
group-theoretically
cuspi-
dal;
group-theoretically
nodal)
if
it
arises
from
an
isomorphism
∼
Π
G
→
Π
H
that
is
group-theoretically
verticial
(respectively,
group-theoretically
cuspidal;
group-theoretically
nodal).
∼
(ii)
We
shall
say
that
an
outer
isomorphism
Π
G
→
Π
H
is
graphic
∼
if
it
arises
from
an
isomorphism
G
→
H.
We
shall
say
that
COMBINATORIAL
ANABELIAN
TOPICS
IV
59
∼
an
isomorphism
Π
G
→
Π
H
is
graphic
if
the
outer
isomorphism
∼
Π
G
→
Π
H
determined
by
it
is
graphic.
Definition
2.8.
Let
G
be
a
semi-graph
of
temperoids
of
HSD-type.
Write
G
for
the
underlying
semi-graph
of
G.
Also,
for
each
z
∈
VCN(G),
write
G
z
for
the
connected
temperoid
of
G
corresponding
to
z.
(i)
Let
H
be
a
sub-semi-graph
of
PSC-type
[cf.
[CbTpI],
Definition
2.2,
(i)]
of
G.
Then
one
may
define
a
semi-graph
of
temperoids
of
HSD-type
G|
H
as
follows
[cf.
Fig.
2
of
[CbTpI]]:
We
take
the
underlying
semi-
graph
of
G|
H
to
be
H;
for
each
vertex
v
(respectively,
edge
e)
of
H,
we
take
the
temperoid
corresponding
to
v
(respectively,
e)
to
be
G
v
(respectively,
G
e
);
for
each
branch
b
of
an
edge
e
of
H
that
abuts
to
a
vertex
v
of
H,
we
take
the
morphism
associated
to
b
to
be
the
morphism
G
e
→
G
v
associated
to
the
branch
of
G
corresponding
to
b.
We
shall
refer
to
G|
H
as
the
semi-graph
of
temperoids
of
HSD-type
obtained
by
restricting
G
to
H.
Thus,
one
has
a
natural
morphism
G|
H
−→
G
of
semi-graphs
of
temperoids.
(ii)
Let
S
⊆
Cusp(G)
be
a
subset
of
Cusp(G)
[cf.
Definition
2.1,
(v)]
which
is
omittable
[cf.
[CbTpI],
Definition
2.4,
(i)]
as
a
subset
of
the
semi-graph
of
anabelioids
of
the
set
of
cusps
Cusp(
G)
of
pro-Primes
PSC-type
G
[cf.
Proposition
2.5,
(iii),
in
the
case
where
Σ
=
Primes]
relative
to
the
natural
identification
Then,
by
eliminating
the
cusps
contained
Cusp(G)
=
Cusp(
G).
in
S,
and,
for
each
vertex
v
of
G,
replacing
the
temperoid
G
v
by
the
temperoid
of
coverings
of
G
v
that
restrict
to
a
trivial
covering
over
the
cusps
contained
in
S
that
abut
to
v,
we
obtain
a
semi-graph
of
temperoids
of
HSD-type
G
•S
[cf.
Fig.
3
of
[CbTpI]].
We
shall
refer
to
G
•S
as
the
partial
compactification
of
G
with
respect
to
S.
(iii)
Let
S
⊆
Node(G)
be
a
subset
of
Node(G)
[cf.
Definition
2.1,
(v)]
such
that
the
semi-graph
obtained
by
removing
the
closed
edges
corresponding
to
the
elements
of
S
from
the
underly-
ing
semi-graph
of
G
is
connected,
i.e.,
in
the
terminology
of
[CbTpI],
Definition
2.5,
(i),
that
is
not
of
separating
type
as
a
of
the
semi-graph
of
anabe-
subset
of
the
set
of
nodes
Node(
G)
lioids
of
pro-Primes
PSC-type
G
[cf.
Proposition
2.5,
(iii),
in
60
YUICHIRO
HOSHI
AND
SHINICHI
MOCHIZUKI
the
case
where
Σ
=
Primes]
relative
to
the
natural
identifica-
Then
one
may
define
a
semi-graph
tion
Node(G)
=
Node(
G).
of
temperoids
of
HSD-type
G
S
as
follows
[cf.
Fig.
4
of
[CbTpI]]:
We
take
the
underlying
semi-
graph
of
G
S
to
be
the
semi-graph
obtained
by
replacing
each
node
e
of
G
contained
in
S
such
that
V(e)
=
{v
1
,
v
2
}
⊆
Vert(G)
[cf.
Definition
2.1,
(v)]
—
where
v
1
,
v
2
are
not
necessarily
dis-
tinct
—
by
two
cusps
that
abut
to
v
1
,
v
2
∈
Vert(G),
respec-
tively,
which
we
think
as
corresponding
to
the
two
branches
of
e.
We
take
the
temperoid
corresponding
to
a
vertex
v
(respectively,
node
e)
of
G
S
to
be
G
v
(respectively,
G
e
).
[Note
that
the
set
of
vertices
(respectively,
nodes)
of
G
S
may
be
naturally
identified
with
Vert(G)
(respectively,
Node(G)
\
S).]
We
take
the
temperoid
corresponding
to
a
cusp
of
G
S
arising
from
a
cusp
e
of
G
to
be
G
e
.
We
take
the
temperoid
corre-
sponding
to
a
cusp
of
G
S
arising
from
a
node
e
of
G
to
be
G
e
.
For
each
branch
b
of
G
S
that
abuts
to
a
vertex
v
of
a
node
e
(respectively,
of
a
cusp
e
that
does
not
arise
from
a
node
of
G),
we
take
the
morphism
associated
to
b
to
be
the
mor-
phism
G
e
→
G
v
associated
to
the
branch
of
G
corresponding
to
b.
For
each
branch
b
of
G
S
that
abuts
to
a
vertex
v
of
a
cusp
of
G
S
that
arises
from
a
node
e
of
G,
we
take
the
mor-
phism
associated
to
b
to
be
the
morphism
G
e
→
G
v
associated
to
the
branch
of
G
corresponding
to
b.
We
shall
refer
to
G
S
as
the
semi-graph
of
temperoids
of
HSD-type
obtained
from
G
by
resolving
S.
Thus,
one
has
a
natural
morphism
G
S
−→
G
of
semi-graphs
of
temperoids.
Remark
2.8.1.
One
verifies
immediately
that
the
operations
of
re-
striction,
partial
compactification,
and
resolution
discussed
in
Defini-
tion
2.8,
(i),
(ii),
(iii),
are
compatible
[in
the
evident
sense]
with
the
corresponding
pro-Σ
operations
—
i.e.,
as
discussed
in
[CbTpI],
Defini-
tion
2.2,
(ii);
[CbTpI],
Definition
2.4,
(ii);
[CbTpI],
Definition
2.5,
(ii)
—
relative
to
the
operation
of
passing
to
the
associated
semi-graph
of
anabelioids
of
pro-Σ
PSC-type
[cf.
Proposition
2.5,
(iii)].
Remark
2.8.2.
We
take
this
opportunity
to
correct
an
unfortunate
misprint
in
[CbTpI],
Definition
2.5,
(ii):
the
phrase
“by
two
cusps
that
abut
to
v
1
,
v
2
∈
Vert(G),
respectively”
of
[CbTpI],
Definition
2.5,
(ii),
COMBINATORIAL
ANABELIAN
TOPICS
IV
61
should
read
“by
two
cusps
that
abut
to
v
1
,
v
2
∈
Vert(G),
respectively,
which
we
think
as
corresponding
to
the
two
branches
of
e”.
Definition
2.9.
In
the
notation
of
Definition
2.8,
let
S
⊆
Node(G)
be
a
subset
of
Node(G)
[cf.
Definition
2.1,
(v)].
Then
we
define
the
semi-graph
of
temperoids
of
HSD-type
G
S
as
follows
[cf.
Fig.
5
of
[CbTpI]]:
def
(i)
We
take
Cusp(G
S
)
=
Cusp(G)
[cf.
Definition
2.1,
(v)].
def
(ii)
We
take
Node(G
S
)
=
Node(G)
\
S
[cf.
Definition
2.1,
(v)].
(iii)
We
take
Vert(G
S
)
[cf.
Definition
2.1,
(v)]
to
be
the
set
of
connected
components
of
the
semi-graph
obtained
from
G
by
omitting
the
edges
e
∈
Edge(G)
\
S
[cf.
Definition
2.1,
(v)].
Alternatively,
one
may
take
Vert(G
S
)
to
be
the
set
of
equiva-
lence
classes
of
elements
of
Vert(G)
with
respect
to
the
equiva-
lence
relation
“∼”
defined
as
follows:
for
v,
w
∈
Vert(G),
v
∼
w
if
either
v
=
w
or
there
exist
n
elements
e
1
,
.
.
.
,
e
n
∈
S
of
S
and
def
n
+
1
vertices
v
0
,
v
1
,
.
.
.
,
v
n
∈
Vert(G)
of
G
such
that
v
0
=
v,
def
v
n
=
w,
and,
for
1
≤
i
≤
n,
it
holds
that
V(e
i
)
=
{v
i−1
,
v
i
}
[cf.
Definition
2.1,
(v)].
(iv)
For
each
branch
b
of
an
edge
e
∈
Edge(G
S
)
(=
Edge(G)
\
S
—
cf.
(i),
(ii))
and
each
vertex
v
∈
Vert(G
S
)
of
G
S
,
b
abuts,
relative
to
G
S
,
to
v
if
b
abuts,
relative
to
G,
to
an
element
of
the
equivalence
class
v
[cf.
(iii)].
(v)
For
each
edge
e
∈
Edge(G
S
)
(=
Edge(G)
\
S
—
cf.
(i),
(ii))
of
G
S
,
we
take
the
temperoid
of
G
S
corresponding
to
e
∈
Edge(G
S
)
to
be
the
temperoid
G
e
.
(vi)
Let
v
∈
Vert(G
S
)
be
a
vertex
of
G
S
.
Then
one
verifies
easily
that
there
exists
a
unique
sub-semi-graph
of
PSC-type
[cf.
[CbTpI],
Definition
2.2,
(i)]
H
v
of
the
underlying
semi-
graph
of
G
whose
set
of
vertices
consists
of
the
elements
of
the
equivalence
class
v
[cf.
(iii)].
Write
def
T
v
=
Node(G|
H
v
)
\
(S
∩
Node(G|
H
v
))
[cf.
Definition
2.8,
(i)].
Then
we
take
the
temperoid
of
G
S
cor-
responding
to
v
∈
Vert(G
S
)
to
be
the
temperoid
B
tp
((G|
H
v
)
T
v
)
[cf.
Definition
2.1,
(vi);
Proposition
2.5,
(i);
Definition
2.8,
(iii)].
(vii)
Let
b
be
a
branch
of
an
edge
e
∈
Edge(G
S
)
(=
Edge(G)
\
S
—
cf.
(i),
(ii))
that
abuts
to
a
vertex
v
∈
Vert(G
S
).
Then
since
b
abuts
to
v,
one
verifies
easily
that
there
exists
a
unique
vertex
w
of
G
which
belongs
to
the
equivalence
class
v
[cf.
(iii)]
such
that
62
YUICHIRO
HOSHI
AND
SHINICHI
MOCHIZUKI
b
abuts
to
w
relative
to
G.
We
take
the
morphism
of
temperoids
associated
to
b,
relative
to
G
S
,
to
be
the
morphism
naturally
determined
by
post-composing
the
morphism
of
temperoids
G
e
→
G
w
corresponding
to
the
branch
b
relative
to
G
with
the
natural
morphism
of
temperoids
G
w
→
B
tp
((G|
H
v
)
T
v
)
[cf.
(vi)].
We
shall
refer
to
this
semi-graph
of
temperoids
of
HSD-type
G
S
as
the
generization
of
G
with
respect
to
S.
Remark
2.9.1.
One
verifies
immediately
that
the
operation
of
gener-
ization
discussed
in
Definition
2.9
is
compatible
[in
the
evident
sense]
with
the
corresponding
pro-Σ
operation
—
i.e.,
as
discussed
in
[CbTpI],
Definition
2.8
—
relative
to
the
operation
of
passing
to
the
associated
semi-graph
of
anabelioids
of
pro-Σ
PSC-type
[cf.
Proposition
2.5,
(iii)].
Remark
2.9.2.
We
take
this
opportunity
to
correct
an
unfortunate
misprint
in
[CbTpI],
Definition
2.8,
(vii):
the
phrase
“equivalent
class”
should
read
“equivalence
class”.
Proposition
2.10
(Specialization
outer
isomorphisms).
Let
G
be
a
semi-graph
of
temperoids
of
HSD-type
and
S
⊆
Node(G)
a
subset
of
Node(G).
Write
Π
G
for
the
fundamental
group
of
G
and
Π
G
S
for
the
fundamental
group
of
the
generization
G
S
of
G
with
respect
to
S
[cf.
Definition
2.9].
Then
there
exists
a
natural
outer
isomorphism
∼
Φ
G
S
:
Π
G
S
−→
Π
G
which
is
functorial,
in
the
evident
sense,
with
respect
to
isomorphisms
of
the
pair
(G,
S)
and
satisfies
the
following
three
conditions:
(a)
Φ
G
S
induces
a
bijection
between
the
set
of
cuspidal
subgroups
[cf.
Definition
2.6,
(i)]
of
Π
G
S
and
the
set
of
cuspidal
sub-
groups
of
Π
G
.
(b)
Φ
G
S
induces
a
bijection
between
the
set
of
nodal
subgroups
[cf.
Definition
2.6,
(i)]
of
Π
G
S
and
the
set
of
nodal
subgroups
of
Π
G
associated
to
the
elements
of
Node(G)
\
S.
(c)
Let
v
∈
Vert(G
S
)
be
a
vertex
of
G
S
;
H
v
,
T
v
as
in
Defini-
tion
2.9,
(vi).
Then
Φ
G
S
induces
a
bijection
between
the
Π
G
S
-
conjugacy
class
of
any
verticial
subgroup
[cf.
Definition
2.6,
(i)]
Π
v
⊆
Π
G
S
of
Π
G
S
associated
to
v
∈
Vert(G
S
)
and
the
Π
G
-
conjugacy
class
of
subgroups
obtained
by
forming
the
image
of
the
outer
homomorphism
Π
(G|
H
v
)
Tv
−→
Π
G
induced
by
the
natural
morphism
(G|
H
v
)
T
v
→
G
[cf.
Defini-
tion
2.8,
(i),
(iii)]
of
semi-graphs
of
temperoids.
COMBINATORIAL
ANABELIAN
TOPICS
IV
63
We
shall
refer
to
this
natural
outer
isomorphism
Φ
G
S
as
the
spe-
cialization
outer
isomorphism
with
respect
to
S.
Proof.
An
outer
isomorphism
that
satisfies
the
three
conditions
in
the
statement
of
Proposition
2.10
may
be
obtained
by
observing
that,
after
sorting
through
the
various
definitions
involved,
an
object
of
B
tp
(G
S
)
amounts
to
the
same
data
as
an
object
of
B
tp
(G).
This
completes
the
proof
of
Proposition
2.10.
Remark
2.10.1.
One
verifies
immediately
that
the
specialization
outer
isomorphism
discussed
in
Proposition
2.10
is
compatible
[in
the
evident
sense]
with
the
corresponding
pro-Σ
outer
isomorphism
—
i.e.,
as
dis-
cussed
in
[CbTpI],
Proposition
2.9
—
relative
to
the
operation
of
pass-
ing
to
the
associated
semi-graph
of
anabelioids
of
pro-Σ
PSC-type
[cf.
Proposition
2.5,
(iii)].
Lemma
2.11
(Infinite
cyclic
coverings).
Let
G
be
a
semi-graph
of
temperoids
of
HSD-type.
Suppose
that
(Vert(G)
,
Node(G)
)
=
(1,
1),
i.e.,
the
semi-graph
of
anabelioids
of
pro-Primes
PSC-type
G
[cf.
Propo-
sition
2.5,
(iii),
in
the
case
where
Σ
=
Primes]
is
cyclically
primitive
[cf.
[CbTpI],
Definition
4.1].
Write
Π
G
for
the
fundamental
group
of
G;
G
for
the
underlying
semi-graph
of
G;
Π
G
(
∼
=
Z)
for
the
discrete
topological
fundamental
group
of
G;
G
∞
→
G
for
the
connected
cover-
ing
of
G
[cf.
Definition
2.1,
(vi)]
corresponding
to
the
natural
surjection
def
Π
G
Π
G
;
Π
G
∞
=
Ker(Π
G
Π
G
).
Then
the
following
hold:
∼
(i)
Fix
an
isomorphism
Π
G
→
Z.
Then
there
exists
a
triple
of
bijections
∼
∼
V
:
Z
−→
Vert(G
∞
),
N
:
Z
−→
Node(G
∞
),
∼
C
:
Z
×
Cusp(G)
−→
Cusp(G
∞
)
[cf.
Definition
2.1,
(v)]
that
satisfies
the
following
properties:
•
The
bijections
are
equivariant
with
respect
to
the
action
∼
of
Π
G
→
Z
on
Z
by
translations
and
the
natural
action
of
Π
G
on
“Vert(−)”,
“Node(−)”,
“Cusp(−)”.
•
The
post-composite
of
C
with
the
natural
map
Cusp(G
∞
)
→
Cusp(G)
coincides
with
the
projection
Z
×
Cusp(G)
→
Cusp(G)
to
the
second
factor.
•
For
each
a
∈
Z,
it
holds
that
E(V
(a))
=
{N
(a),
N
(a
+
1)}
{
C(a,
z)
|
z
∈
Cusp(G)
}
[cf.
Definition
2.1,
(v)].
Moreover,
such
a
triple
of
bijections
is
unique,
up
to
post-
composition
with
the
automorphisms
of
“Vert(−)”,
“Node(−)”,
“Cusp(−)”
determined
by
the
action
of
a
[single!]
element
of
Π
G
.
64
YUICHIRO
HOSHI
AND
SHINICHI
MOCHIZUKI
(ii)
Let
a
≤
b
be
integers.
Write
G
[a,b]
for
the
[uniquely
deter-
mined]
sub-semi-graph
of
PSC-type
[cf.
[CbTpI],
Definition
2.2,
(i)]
of
the
underlying
semi-graph
of
G
∞
whose
set
of
ver-
tices
is
equal
to
{V
(a),
V
(a+1),
.
.
.
,
V
(b)}
[cf.
(i)].
Also,
write
G
[a,b]
for
the
semi-graph
of
temperoids
obtained
by
restricting
G
∞
to
G
[a,b]
[in
the
evident
sense
—
cf.
also
the
procedure
dis-
cussed
in
Definition
2.8,
(i)].
Then
G
[a,b]
is
a
semi-graph
of
temperoids
of
HSD-type.
(iii)
Let
a
≤
b
be
integers.
For
an
integer
c
such
that
a
≤
c
≤
b
(respectively,
a
+
1
≤
c
≤
b),
let
Π
V
(c)
⊆
Π
G
[a,b]
(respectively,
Π
N
(c)
⊆
Π
G
[a,b]
)
be
a
verticial
(respectively,
nodal)
subgroup
of
Π
G
[a,b]
associated
to
V
(c)
∈
Vert(G
[a,b]
)
(respectively,
N
(c)
∈
Node(G
[a,b]
))
[cf.
(i),
(ii)]
such
that,
for
a
+
1
≤
c
≤
b,
it
holds
that
Π
N
(c)
⊆
Π
V
(c−1)
∩
Π
V
(c)
.
Then
the
inclusions
Π
V
(c)
,
Π
N
(c)
→
Π
G
[a,b]
determine
an
isomorphism
lim
Π
V
(a)
←
Π
N
(a+1)
→
Π
V
(a+1)
←
·
·
·
→
Π
V
(b−1)
←
Π
N
(b)
→
Π
V
(b)
−→
∼
−→
Π
G
[a,b]
—
where
lim
denotes
the
inductive
limit
in
the
category
of
−→
groups.
(iv)
Let
a
≤
b
be
integers.
Then
the
composite
G
[a,b]
→
G
∞
→
G
determines
an
outer
injection
Π
G
[a,b]
→
Π
G
.
Moreover,
the
image
of
this
outer
injection
is
contained
in
the
normal
subgroup
Π
G
∞
⊆
Π
G
.
(v)
There
exists
a
collection
{D
[−a,a]
}
1≤a∈Z
of
subgroups
D
[−a,a]
⊆
Π
G
∞
indexed
by
the
positive
integers
which
satisfy
the
following
properties:
•
D
[−a,a]
⊆
Π
G
∞
belongs
to
the
Π
G
-conjugacy
class
[of
sub-
groups
of
Π
G
]
obtained
by
forming
the
image
of
the
outer
injection
Π
G
[−a,a]
→
Π
G
of
(iv).
•
D
[−a,a]
⊆
D
[−a−1,a+1]
.
•
The
inclusions
D
[−a,a]
→
Π
G
[where
a
ranges
over
the
positive
integers]
determine
an
isomorphism
∼
lim
D
[−1,1]
→
D
[−2,2]
→
D
[−3,3]
→
·
·
·
−→
Π
G
∞
−→
—
where
lim
denotes
the
inductive
limit
in
the
category
of
−→
groups.
(vi)
In
the
situation
of
(v),
since
Π
G
injects
into
its
pro-l
com-
pletion
for
any
l
∈
Primes
[cf.
Remark
2.5.1],
let
us
regard
subgroups
of
Π
G
as
subgroups
of
the
pro-Σ
completion
Π
Σ
G
of
Σ
Π
G
.
Let
a
be
a
positive
integer.
Write
D
[−a,a]
⊆
Π
G
for
the
closure
of
D
[−a,a]
in
Π
Σ
∈
Π
Σ
G
.
Let
γ
G
.
Suppose
that
D
[a,−a]
∩
COMBINATORIAL
ANABELIAN
TOPICS
IV
65
γ
·
D
[a,−a]
·
γ
−1
=
{1}.
Then
the
image
of
γ
∈
Π
Σ
G
in
the
pro-Σ
Σ
Σ
completion
Π
G
of
Π
G
is
contained
in
Π
G
⊆
Π
G
.
(vii)
In
the
situation
of
(vi),
suppose,
moreover,
that
γ
is
contained
Σ
of
Π
in
Π
.
Then
γ
∈
D
[a,−a]
.
in
the
closure
Π
G
∞
⊆
Π
Σ
G
∞
G
G
Proof.
Assertions
(i),
(ii)
follow
immediately
from
the
various
defini-
tions
involved.
Assertion
(iii)
follows
immediately
from
a
similar
argu-
ment
to
the
argument
applied
in
the
proof
of
[CmbCsp],
Proposition
1.5,
(iii).
Next,
we
verify
assertion
(iv).
The
injectivity
portion
of
asser-
tion
(iv)
follows
immediately
—
by
considering
a
suitable
finite
étale
subcovering
of
G
∞
→
G
and
applying
a
suitable
specialization
outer
isomorphism
[cf.
Proposition
2.10]
—
from
Proposition
2.5,
(iv).
The
remainder
of
assertion
(iv)
follows
immediately
from
the
various
defi-
nitions
involved.
This
completes
the
proof
of
assertion
(iv).
Assertion
(v)
follows
immediately
from
assertion
(iii).
Next,
we
verify
assertion
(vi).
Write
G
Σ
for
the
semi-graph
of
an-
abelioids
of
pro-Σ
PSC-type
determined
by
G
[cf.
Proposition
2.5,
(iii)],
G
Σ
→
G
Σ
for
the
universal
covering
of
the
semi-graph
of
anabe-
lioids
of
pro-Σ
PSC-type
G
corresponding
to
[the
torsion-free
group]
Σ
for
the
Π
Σ
G
[cf.
Proposition
2.5,
(iii);
[MT],
Remark
1.2.2],
and
G
underlying
pro-semi-graph
of
G
Σ
.
Then
it
follows
immediately
—
i.e.,
by
considering
a
suitable
finite
étale
subcovering
of
G
∞
→
G
and
applying
a
suitable
specialization
outer
isomorphism
[cf.
Propo-
sition
2.10]
—
from
[NodNon],
Lemma
1.9,
(ii),
that
our
assumption
that
D
[a,−a]
∩
γ
·
D
[a,−a]
·
γ
−1
=
{1}
implies
that
the
respective
sub-
Σ
determined
by
D
[a,−a]
,
γ
pro-semi-graphs
of
G
·
D
[a,−a]
·
γ
−1
⊆
Π
Σ
G
[cf.
Proposition
2.5,
(v)]
either
contain
a
common
pro-vertex
or
may
be
joined
to
one
another
by
a
single
pro-edge.
But
this
implies
that
γ
maps
G
[−a,a]
to
some
Π
G
-translate
of
G
[−a,a]
,
hence,
in
particular,
Σ
Σ
that
the
image
of
γ
∈
Π
Σ
G
in
Π
G
is
contained
in
Π
G
⊆
Π
G
,
as
desired.
This
completes
the
proof
of
assertion
(vi).
Assertion
(vii)
follows
im-
mediately
—
i.e.,
by
considering
a
suitable
finite
étale
subcovering
of
G
∞
→
G
and
applying
a
suitable
specialization
outer
isomorphism
[cf.
Proposition
2.10]
—
from
the
commensurable
terminality
[cf.
[CmbGC],
Proposition
1.2,
(ii)]
of
D
[a,−a]
in
a
suitable
open
subgroup
of
Π
Σ
G
con-
taining
Π
G
∞
[cf.
also
[NodNon],
Lemma
1.9,
(ii)].
This
completes
the
proof
of
Lemma
2.11.
The
content
of
the
following
lemma
is
entirely
elementary
and
well-
known.
Lemma
2.12
(Action
of
the
symplectic
group).
Let
g
be
a
pos-
itive
integer.
For
each
positive
integer
n
and
v
=
(v
1
,
.
.
.
,
v
n
)
∈
Z
⊕n
,
write
vol(v)
∈
Z
for
the
[uniquely
determined]
nonnegative
integer
that
66
YUICHIRO
HOSHI
AND
SHINICHI
MOCHIZUKI
generates
the
ideal
Z
·
v
1
+
·
·
·
+
Z
·
v
n
⊆
Z;
M
n
(Z)
for
the
set
of
n
by
n
matrices
with
coefficients
in
Z;
GL
n
(Z)
⊆
M
n
(Z)
for
the
group
of
matrices
A
∈
M
n
(Z)
such
that
det(A)
∈
{1,
−1};
Sp
2g
(Z)
⊆
GL
2g
(Z)
for
the
subgroup
of
2g
by
2g
symplectic
matrices,
i.e.,
B
∈
GL
2g
(Z)
such
that
0
1
t
0
1
B
·
·
B
=
.
−1
0
−1
0
[Note
that
one
verifies
immediately
that,
for
every
A
∈
GL
n
(Z),
it
holds
that
vol(v)
=
vol(vA).]
Then
the
following
hold:
(i)
Let
v
=
(v
1
,
.
.
.
,
v
g
)
∈
Z
⊕g
.
Then
there
exists
an
invertible
g−1
matrix
A
∈
GL
g
(Z)
such
that
vA
=
(vol(v),
0,
.
.
.
,
0).
(ii)
Let
v
=
(v
1
,
.
.
.
,
v
2g
)
∈
Z
⊕2g
.
Then
there
exists
a
symplectic
2g−1
matrix
B
∈
Sp
2g
(Z)
such
that
vB
=
(vol(v),
0,
.
.
.
,
0).
(iii)
Let
N
⊆
Z
⊕2g
be
a
submodule
of
Z
⊕2g
and
v
∈
Z
⊕2g
.
Suppose
that
N
=
{0}.
Then
there
exist
a
nonzero
integer
n
∈
Z
\
{0}
and
a
symplectic
matrix
B
∈
Sp
2g
(Z)
such
that
n
·
vB
∈
N
.
(iv)
Let
N
⊆
Z
⊕2g
be
a
submodule
of
Z
⊕2g
and
π
:
Z
⊕2g
Z
a
surjection.
Suppose
that
N
is
of
infinite
index
in
Z
⊕2g
.
Then
there
exists
a
symplectic
matrix
B
∈
Sp
2g
(Z)
such
that
N
·
B
⊆
Ker(π).
Proof.
First,
we
verify
assertion
(i).
Let
us
first
observe
that
if
v
=
0
[i.e.,
vol(v)
=
0],
then
assertion
(i)
is
immediate.
Thus,
to
verify
assertion
(i),
we
may
assume
without
loss
of
generality
that
v
=
0.
In
particular,
to
verify
assertion
(i),
by
replacing
v
by
vol(v)
−1
·
v
∈
Z
⊕g
,
we
may
assume
without
loss
of
generality
that
vol(v)
=
1.
On
the
other
hand,
since
vol(v)
=
1,
one
verifies
immediately
that
Z
⊕g
/(Z
·
v)
is
a
free
Z-module
of
rank
g
−
1,
hence
that
there
exists
an
injection
∼
Z
⊕g−1
→
Z
⊕g
that
induces
an
isomorphism
(Z
·
v)
⊕
Z
⊕g−1
→
Z
⊕g
.
This
completes
the
proof
of
assertion
(i).
Next,
we
verify
assertion
(ii).
Since
[one
verifies
easily
that]
Sp
2
(Z)
=
SL
2
(Z)
=
{
B
∈
GL
2
(Z)
|
det(B)
=
1
},
assertion
(ii)
in
the
case
where
g
=
1
follows
immediately
from
assertion
(i)
[in
the
case
where
we
take
“g”
in
assertion
(i)
to
be
2],
together
with
the
[easily
verified]
fact
that
a
b
a
−b
a
b
det
,
det
=
{1,
−1}
for
every
∈
GL
2
(Z).
c
d
c
−d
c
d
For
i
∈
{1,
.
.
.
,
g},
write
M
i
for
the
submodule
of
Z
⊕2g
generated
by
(0,
.
.
.
,
0,
1,
0,
.
.
.
,
0),
(0,
.
.
.
,
0,
1,
0,
.
.
.
,
0)
∈
Z
⊕2g
—
where
the
“1’s”
lie,
respectively,
in
the
i-th
and
(g
+
i)-th
compo-
nents.
Then,
by
applying
assertion
(ii)
in
the
case
where
g
=
1
[already
verified
above]
to
the
M
i
’s,
we
conclude
that,
to
complete
the
verifi-
cation
of
assertion
(ii),
we
may
assume
without
loss
of
generality
that
COMBINATORIAL
ANABELIAN
TOPICS
IV
67
def
v
i
=
0
for
every
g
+
1
≤
i
≤
2g.
Write
v
≤g
=
(v
1
,
.
.
.
,
v
g
)
∈
Z
⊕g
.
Then
let
us
observe
that
it
follows
from
assertion
(i)
that
there
exists
an
invertible
matrix
A
∈
GL
g
(Z)
such
that
v
≤g
A
=
(vol(v
≤g
),
0,
.
.
.
,
0)
=
(vol(v),
0,
.
.
.
,
0).
Thus,
assertion
(ii)
follows
immediately
from
the
[easily
verified]
fact
that
A
0
∈
Sp
2g
(Z).
0
t
A
−1
This
completes
the
proof
of
assertion
(ii).
Assertion
(iii)
follows
immediately
from
assertion
(ii).
Assertion
(iv)
⊕2g
follows
immediately
—
by
applying
the
self-duality
with
respect
of
Z
0
1
to
the
symplectic
form
determined
by
—
from
assertion
(iii).
−1
0
This
completes
the
proof
of
Lemma
2.12.
Lemma
2.13
(Automorphisms
of
surface
groups).
Let
g
be
a
positive
integer,
Π
the
topological
fundamental
group
of
a
connected
orientable
compact
topological
surface
of
genus
g,
π
:
Π
Z
a
surjec-
tion,
and
J
⊆
Π
a
subgroup
of
Π
such
that
the
image
of
J
in
Π
ab
is
of
infinite
index
in
Π
ab
.
[For
example,
this
will
be
the
case
if
J
is
generated
by
2g
−
1
elements.]
Then
there
exists
an
automorphism
σ
of
Π
such
that
σ(J)
⊆
Ker(π).
def
Proof.
Write
H
=
Hom(Π,
Z)
=
Hom
Z
(Π
ab
,
Z).
Let
us
fix
isomor-
∼
∼
phisms
H
→
Z
⊕2g
and
H
2
(Π,
Z)
→
Z.
Then
it
follows
from
the
well-known
theory
of
Poincaré
duality
that
the
cup
product
in
group
cohomology
H
×
H
=
H
1
(Π,
Z)
×
H
1
(Π,
Z)
−→
H
2
(Π,
Z)
∼
=
Z
determines
a
perfect
pairing
on
H;
moreover,
if
we
write
Aut
PD
(H)
⊆
∼
Aut(H)
(
→
GL
2g
(Z)
—
cf.
the
notation
of
Lemma
2.12)
for
the
sub-
group
of
automorphisms
of
H
that
are
compatible
with
this
perfect
∼
pairing,
then
—
by
replacing
the
isomorphism
H
→
Z
⊕2g
by
a
suit-
∼
able
isomorphism
if
necessary
—
the
isomorphism
Aut(H)
→
GL
2g
(Z)
∼
determines
an
isomorphism
Aut
PD
(H)
→
Sp
2g
(Z)
[cf.
the
notation
of
Lemma
2.12].
On
the
other
hand,
recall
[cf.,
e.g.,
the
discussion
preced-
ing
[DM],
Theorem
5.13]
that
the
natural
homomorphism
Aut(Π)
→
Aut(H)
determines
a
surjection
Aut(Π)
Aut
PD
(H)
(⊆
Aut(H)).
Thus,
Lemma
2.13
follows
immediately
from
Lemma
2.12,
(iv).
This
completes
the
proof
of
Lemma
2.13.
Lemma
2.14
(Finitely
generated
subgroups
of
surface
groups).
Let
G
be
a
semi-graph
of
temperoids
of
HSD-type
and
J
⊆
Π
G
a
finitely
68
YUICHIRO
HOSHI
AND
SHINICHI
MOCHIZUKI
generated
subgroup
of
the
fundamental
group
Π
G
of
G.
Then
the
fol-
lowing
hold:
(i)
Suppose
that
Cusp(G)
=
∅.
Then
there
exist
a
subgroup
F
⊆
Π
G
of
finite
index
and
a
surjection
F
J
such
that
J
⊆
F
,
and,
moreover,
the
restriction
of
the
surjection
F
J
to
J
⊆
F
is
the
identity
automorphism
of
J.
(ii)
Suppose
that
(Vert(G)
,
Cusp(G)
,
Node(G)
)
=
(1,
0,
1).
Thus,
since
we
are
in
the
situation
of
Lemma
2.11,
we
shall
apply
the
notational
conventions
established
in
Lemma
2.11.
Suppose
ab
that
the
image
of
J
in
Π
ab
G
is
of
infinite
index
in
Π
G
.
[For
example,
this
will
be
the
case
if
J
is
generated
by
rank
Z
(Π
ab
G
)−1
elements.]
Then
there
exists
an
automorphism
σ
∈
Aut(Π
G
)
of
Π
G
such
that
σ(J)
⊆
Π
G
∞
.
(iii)
In
the
situation
of
(ii),
suppose,
moreover,
that
J
⊆
Π
G
∞
.
Then
there
exists
a
positive
integer
a
∈
Z
such
that
J
⊆
D
[−a,a]
[cf.
Lemma
2.11,
(v)].
Proof.
Assertion
(i)
follows
from
[SemiAn],
Corollary
1.6,
(ii),
together
with
the
fact
that
Π
G
is
a
finitely
generated
free
group
[cf.
Remark
2.5.1].
Assertion
(ii)
follows
from
Lemma
2.13.
Assertion
(iii)
follows
from
Lemma
2.11,
(v),
together
with
our
assumption
that
J
is
finitely
gen-
erated.
This
completes
the
proof
of
Lemma
2.14.
Theorem
2.15
(Profinite
conjugates
of
finitely
generated
Primes-
compatible
subgroups).
Let
Π
be
the
topological
fundamental
group
of
a
compact
orientable
hyperbolic
topological
surface
with
compact
bound-
ary
[cf.
Remark
2.5.1]
and
H,
J
⊆
Π
subgroups.
Since
Π
injects
into
its
pro-l
completion
for
any
l
∈
Primes
[cf.,
e.g.,
[RZ],
Proposition
3.3.15;
[Prs],
Theorem
1.7],
let
us
regard
subgroups
of
Π
as
subgroups
of
Π.
Write
H,
J
⊆
Π
for
the
closures
of
the
profinite
completion
Π
respectively.
Suppose
that
the
following
conditions
are
of
H,
J
in
Π,
satisfied:
(a)
The
subgroups
H
and
J
are
finitely
generated.
(b)
If
J
is
of
infinite
index
in
Π,
then
J
is
of
infinite
index
in
Π.
[Here,
we
note
that
condition
(b)
is
automatically
satisfied
whenever
Π
is
free
—
cf.
[SemiAn],
Corollary
1.6,
(ii).]
Then
the
following
hold:
(i)
It
holds
that
J
=
J
∩
Π.
such
that
(ii)
Suppose
that
there
exists
an
element
γ
∈
Π
H
⊆
γ
·
J
·
γ
−1
.
Then
there
exists
an
element
δ
∈
Π
such
that
H
⊆
δ
·
J
·
δ
−1
.
COMBINATORIAL
ANABELIAN
TOPICS
IV
69
Proof.
Let
us
first
observe
that,
to
verify
Theorem
2.15,
we
may
assume
without
loss
of
generality
that
Π
is
the
fundamental
group
Π
G
of
a
semi-graph
of
temperoids
of
HSD-type
G
[cf.
Definition
2.3].
Next,
we
claim
that
the
following
assertion
holds:
Claim
2.15.A:
Theorem
2.15
holds
in
the
case
where
J
is
of
finite
index
in
Π
G
.
Indeed,
write
N
⊆
Π
G
for
the
normal
subgroup
of
Π
G
obtained
by
forming
the
intersection
of
all
Π
G
-conjugates
of
J.
Then
since
J
is
of
finite
index
in
Π
G
,
it
is
immediate
that
N
is
of
finite
index
in
Π
G
.
Thus,
by
considering
the
images
in
Π
G
/N
of
the
various
groups
involved,
one
verifies
immediately
that
Theorem
2.15
holds
in
the
case
where
J
is
of
finite
index
in
Π
G
.
This
completes
the
proof
of
Claim
2.15.A.
Thus,
in
the
remainder
of
the
proof
of
Theorem
2.15,
we
may
assume
without
loss
of
generality
that
J
is
of
infinite
index
in
Π
G
,
which
implies
that
G
[cf.
condition
(b)].
J
is
of
infinite
index
in
Π
Next,
we
claim
that
the
following
assertion
holds:
Claim
2.15.B:
Let
F
⊆
Π
G
be
a
subgroup
of
finite
index
such
that
J
⊆
F
.
Suppose
that
the
assertion
obtained
by
replacing
Π
G
in
assertion
(i)
by
F
holds.
Then
assertion
(i)
holds,
and,
in
the
situation
of
as-
sertion
(ii),
there
exists
a
Π
G
-conjugate
of
H
that
is
contained
in
F
.
If,
moreover,
the
assertion
obtained
by
replacing
Π
G
in
assertion
(ii)
by
F
holds,
then
as-
sertion
(ii)
holds.
Indeed,
let
us
first
observe
that
since
the
natural
inclusion
F
→
Π
G
is
Primes-compatible
[cf.
the
discussion
entitled
“Groups”
in
§0],
the
profinite
completion
F
of
F
may
be
identified
with
the
closure
F
of
F
G
.
In
particular,
the
closure
of
J
in
F
is
naturally
isomorphic
to
the
in
Π
G
.
Thus,
it
follows
from
Claim
2.15.A
applied
to
F
closure
J
of
J
in
Π
that
the
assertion
obtained
by
replacing
Π
G
in
assertion
(i)
by
F
implies
assertion
(i).
Next,
let
us
observe
that
in
the
situation
of
assertion
(ii),
G
,
by
replacing
H
by
a
since
[one
verifies
immediately
that]
Π
G
·
F
=
Π
suitable
Π
G
-conjugate
of
H,
we
may
assume
without
loss
of
generality
that
γ
∈
F
.
In
particular,
since
H
⊆
γ
·
J
·
γ
−1
⊆
γ
·
F
·
γ
−1
=
F
,
it
follows
that
H
⊆
F
∩
Π
G
=
F
[cf.
Claim
2.15.A].
Thus,
one
verifies
easily
that
the
assertion
obtained
by
replacing
Π
G
in
assertion
(ii)
by
F
implies
assertion
(ii).
This
completes
the
proof
of
Claim
2.15.B.
Next,
we
verify
Theorem
2.15
in
the
case
where
Cusp(G)
=
∅.
Suppose
that
Cusp(G)
=
∅.
Then
it
follows
from
Lemma
2.14,
(i),
that
there
exist
a
subgroup
F
⊆
Π
G
of
finite
index
and
a
surjection
π
:
F
J
such
that
J
⊆
F
,
and,
moreover,
the
restriction
of
π
to
J
⊆
F
is
the
identity
automorphism
of
J.
Now
it
follows
immediately
from
Claim
2.15.B
that,
by
replacing
Π
G
by
F
,
we
may
assume
without
loss
of
generality
that
Π
G
=
F
.
Next,
let
us
observe
that
since
[it
is
70
YUICHIRO
HOSHI
AND
SHINICHI
MOCHIZUKI
immediate
that]
J
⊆
J
∩
Π
G
,
to
complete
the
verification
of
assertion
(i)
in
the
case
where
Cusp(G)
=
∅,
it
suffices
to
verify
that
J
∩
Π
G
⊆
J.
Moreover,
since
J
⊆
J
∩
Π
G
(⊆
J),
it
follows
immediately
from
the
G
J
for
the
surjection
in-
equality
π
|
J
=
id
J
[where
we
write
π
:
Π
duced
by
π]
that,
to
verify
the
inclusion
J
∩
Π
G
⊆
J,
it
suffices
to
verify
that
π
(J
∩
Π
G
)
⊆
π
(J).
On
the
other
hand,
one
verifies
easily
that
π
(J
∩
Π
G
)
⊆
π
(Π
G
)
=
J
=
π
(J),
as
desired.
This
completes
the
proof
of
assertion
(i)
in
the
case
where
Cusp(G)
=
∅.
Next,
to
verify
assertion
(ii)
in
the
case
where
Cusp(G)
=
∅,
let
us
observe
that,
by
replacing
γ
by
γ
·
π
(
γ
−1
),
we
may
assume
without
loss
of
generality
that
γ
∈
Ker(
π
).
Now
we
claim
that
the
following
assertion
holds:
Claim
2.15.C:
It
holds
that
H
⊆
γ
·
J
·
γ
−1
.
,
J
⊆
J,
it
follows
Indeed,
since
[one
verifies
easily
that]
γ
−1
·
H
·
γ
immediately
from
the
equality
π
|
J
=
id
J
that,
to
verify
Claim
2.15.C,
it
suffices
to
verify
that
π
(
γ
−1
·
H
·
γ
)
⊆
π
(J).
On
the
other
hand,
since
γ
∈
Ker(
π
),
it
holds
that
)
=
π
(H)
⊆
π
(Π
G
)
=
J
=
π
(J),
π
(
γ
−1
·
H
·
γ
as
desired.
This
completes
the
proof
of
Claim
2.15.C.
In
particular,
it
follows
immediately
from
[IUTeichI],
Theorem
2.6
[i.e.,
in
essence,
the
argument
given
in
the
proof
of
[André],
Lemma
3.2.1],
that
there
−1
·
H
·
γ
⊆
J.
This
exists
an
element
δ
∈
Π
G
such
that
δ
−1
·
H
·
δ
=
γ
completes
the
proof
of
assertion
(ii)
in
the
case
where
Cusp(G)
=
∅,
hence
also
of
Theorem
2.15
in
the
case
where
Cusp(G)
=
∅.
Next,
we
verify
Theorem
2.15
in
the
case
where
Cusp(G)
=
∅.
Sup-
pose
that
Cusp(G)
=
∅.
First,
we
observe
that
since
J
is
of
infinite
G
,
it
follows
immediately
that
[Π
G
:
J
·N
]
→
+∞
as
N
ranges
index
in
Π
over
the
normal
subgroups
of
Π
G
of
finite
index,
hence
[cf.
Claim
2.15.B;
the
fact
that
J
is
finitely
generated]
that,
by
replacing
Π
G
by
a
suitable
subgroup
of
finite
index
in
Π
G
that
contains
J,
we
may
assume
without
ab
loss
of
generality
that
the
image
of
J
in
Π
ab
G
is
of
infinite
index
in
Π
G
[cf.
Remark
2.5.1].
Moreover,
by
considering
suitable
specialization
outer
isomorphisms
[cf.
Proposition
2.10],
we
may
assume
without
loss
of
generality
that
the
equality
(Vert(G)
,
Cusp(G)
,
Node(G)
)
=
(1,
0,
1)
holds.
Thus,
since
we
are
in
the
situation
of
Lemma
2.11,
we
shall
apply
the
notational
conventions
established
in
Lemma
2.11.
More-
over,
it
follows
from
Lemma
2.14,
(ii),
that,
by
considering
a
suitable
automorphism
of
Π
G
,
we
may
assume
without
loss
of
generality
that
J
⊆
Π
G
∞
.
Thus,
it
follows
from
Lemma
2.14,
(iii),
that
there
exists
a
positive
integer
a
∈
Z
such
that
J
⊆
D
[−a,a]
⊆
Π
G
∞
.
COMBINATORIAL
ANABELIAN
TOPICS
IV
71
∼
Next,
let
us
observe
that
since
Π
G
/Π
G
∞
→
Π
G
(
∼
=
Z)
injects
into
its
profinite
completion,
it
follows
that
J
∩
Π
G
⊆
Π
G
∞
.
In
particular,
by
applying
Lemma
2.14,
(iii),
we
conclude
that,
for
any
given
fixed
element
α
∈
J
∩
Π
G
,
we
may
assume,
by
possibly
enlarging
a,
that
α
∈
D
[−a,a]
.
Next,
let
us
observe
—
i.e.,
by
considering
a
suitable
finite
étale
subcovering
of
G
∞
→
G
and
applying
a
suitable
specialization
outer
isomorphism
[cf.
Proposition
2.10]
—
that
the
natural
inclusion
D
[−a,a]
→
Π
G
is
Primes-compatible
[cf.
Proposition
2.5,
(iv)].
In
par-
ticular,
by
replacing
G
by
G
[−a,a]
[cf.
Lemma
2.11,
(ii)],
we
conclude
that
assertion
(i)
in
the
case
where
Cusp(G)
=
∅
follows
from
asser-
tion
(i)
in
the
case
where
Cusp(G)
=
∅
[already
verified
above].
This
completes
the
proof
of
assertion
(i)
in
the
case
where
Cusp(G)
=
∅.
Finally,
to
verify
assertion
(ii)
in
the
case
where
Cusp(G)
=
∅,
let
us
observe
that
if
H
=
{1},
then
assertion
(ii)
is
immediate.
Thus,
we
may
assume
without
loss
of
generality
that
H
=
{1}.
Next,
let
us
observe
∼
that
since
J
⊆
D
[−a,a]
⊆
Π
G
∞
,
and
Π
G
/Π
G
∞
→
Π
G
(
∼
=
Z)
injects
into
its
profinite
completion,
one
verifies
immediately
that
H
⊆
Π
G
∞
.
Thus,
since
H
⊆
Π
G
∞
is
finitely
generated,
it
follows
from
Lemma
2.14,
(iii),
that,
by
possibly
enlarging
a,
we
may
assume
without
loss
of
generality
·
J
·
γ
−1
⊆
that
H
⊆
D
[−a,a]
.
Since,
moreover,
{1}
=
H
⊆
D
[−a,a]
∩
γ
D
[−a,a]
∩
γ
·D
[−a,a]
·
γ
−1
,
it
follows
from
Lemma
2.11,
(vi),
that
the
image
G
of
Π
G
is
contained
in
Π
G
⊆
Π
G
,
G
in
the
profinite
completion
Π
of
γ
∈
Π
which
thus
implies
that
there
exists
an
element
γ
∈
Π
G
such
that
γ
γ
∈
Π
G
∞
.
In
particular,
by
replacing
H
by
γ
·
H
·
(γ
)
−1
and
possibly
enlarging
a,
we
may
assume
without
loss
of
generality
that
γ
∈
Π
G
∞
.
Thus,
again
by
applying
the
fact
that
{1}
=
D
[−a,a]
∩
γ
·
D
[−a,a]
·
γ
−1
,
we
conclude
from
Lemma
2.11,
(vii),
that
γ
∈
D
[−a,a]
.
In
particular,
since,
as
discussed
above
[cf.
the
discussion
immediately
preceding
the
proof
of
assertion
(i)
in
the
case
where
Cusp(G)
=
∅],
the
natural
inclusion
D
[−a,a]
→
Π
G
is
Primes-compatible,
by
replacing
G
by
G
[−a,a]
,
we
conclude
that
assertion
(ii)
in
the
case
where
Cusp(G)
=
∅
follows
from
assertion
(ii)
in
the
case
where
Cusp(G)
=
∅
[already
verified
above].
This
completes
the
proof
of
assertion
(ii)
in
the
case
where
Cusp(G)
=
∅,
hence
also
of
Theorem
2.15.
Remark
2.15.1.
In
passing,
we
observe
that
the
analogue
of
Theo-
rem
2.15
for
arbitrary
Σ
=
Primes
is
false.
Indeed,
if,
in
the
statement
of
Theorem
2.15,
one
replaces
“Π”
by
the
group
Z,
then
it
is
easy
to
construct
counterexamples
to
assertions
(i),
(ii).
One
may
then
obtain
counterexamples
in
the
case
of
the
original
“Π”
by
considering
the
case
where
the
original
“Π”
is
the
fundamental
group
Π
G
of
a
semi-graph
of
temperoids
of
HSD-type
G
such
that
Edge(G)
=
∅
and
considering
suitable
edge-like
subgroups
[i.e.,
isomorphic
to
Z!]
of
Π
G
.
72
YUICHIRO
HOSHI
AND
SHINICHI
MOCHIZUKI
Lemma
2.16
(VCN-subgroups
of
infinite
index).
Let
G
be
a
semi-
graph
of
anabelioids
of
pro-Σ
PSC-type
(respectively,
of
temperoids
of
def
def
HSD-type).
Write
J
=
Π
Σ
G
(respectively,
J
=
Π
G
)
for
the
[pro-Σ
(respectively,
discrete)]
fundamental
group
of
G.
Let
H
⊆
J
be
a
VCN-
subgroup
of
J.
Consider
the
following
two
[mutually
exclusive]
condi-
tions:
(1)
H
=
J.
(2)
H
is
of
infinite
index
in
J.
Then
we
have
equivalences
(1)
⇐⇒
(1
);
(2)
⇐⇒
(2
)
with
the
following
two
conditions:
(1
)
H
is
verticial,
and
Node(G)
=
∅.
(2
)
Either
H
is
edge-like,
or
Node(G)
=
∅.
Proof.
The
implication
(1
)
⇒
(1)
follows
immediately
from
the
various
definitions
involved.
Thus,
one
verifies
immediately
[by
considering
suitable
contrapositive
versions
of
the
various
implications
involved]
that,
to
complete
the
verification
of
Lemma
2.16,
it
suffices
to
verify
the
implication
(2
)
⇒
(2).
To
this
end,
let
us
observe
that
if
H
is
edge-
like,
then
since
H
is
abelian,
and
every
closed
subgroup
of
J
of
finite
index
is
center-free
[cf.,
e.g.,
Remark
2.5.1;
[CmbGC],
Remark
1.1.3],
we
conclude
that
H
is
of
infinite
index
in
J.
Thus,
we
may
assume
without
loss
of
generality
that
H
is
verticial
and
Node(G)
=
∅.
Now
since
Node(G)
=
∅,
it
follows
from
a
similar
argument
to
the
argument
in
the
discussion
entitled
“Curves”
in
[AbsTpII],
§0,
that,
by
replacing
G
by
a
suitable
connected
finite
étale
covering
of
G,
we
may
assume
without
loss
of
generality
that
the
underlying
semi-graph
of
G
is
loop-
ample
[cf.
the
discussion
entitled
“Semi-graphs”
in
[AbsTpII],
§0].
In
particular,
since
[one
verifies
easily
that]
the
abelianization
of
the
[pro-
Σ
completion
of
the]
topological
fundamental
group
of
a
noncontractible
semi-graph
is
infinite,
the
image
of
H
in
the
abelianization
of
J
is
of
infinite
index,
which
thus
implies
that
H
is
of
infinite
index
in
J,
as
desired.
This
completes
the
proof
of
Lemma
2.16.
Corollary
2.17
(Profinite
conjugates
of
VCN-subgroups).
Let
G
and
H
be
semi-graphs
of
temperoids
of
HSD-type.
Write
Π
G
,
Π
H
for
the
respective
fundamental
groups
of
G,
H.
Thus,
we
obtain
a
semi-graph
of
[cf.
Proposition
2.5,
(iii),
in
the
anabelioids
of
pro-Primes
PSC-type
H
case
where
Σ
=
Primes].
Let
z
G
∈
VCN(G),
z
H
∈
VCN(H),
Π
z
G
⊆
Π
G
a
VCN-subgroup
of
Π
G
associated
to
z
G
∈
VCN(G),
Π
z
H
⊆
Π
H
a
VCN-
subgroup
of
Π
H
associated
to
z
H
∈
VCN(H),
∼
α
:
Π
G
−→
Π
H
COMBINATORIAL
ANABELIAN
TOPICS
IV
73
an
isomorphism
of
groups,
and
γ
∈
Π
H
an
element
of
the
[profinite]
Let
us
fix
an
injection
Π
H
→
Π
such
fundamental
group
Π
H
of
H.
H
that
the
induced
outer
injection
is
the
outer
injection
of
Proposition
2.5,
(iii),
and
regard
subgroups
of
Π
H
as
subgroups
of
Π
H
by
means
of
this
fixed
injection.
Write
Π
z
H
⊆
Π
H
for
the
closure
of
Π
z
H
in
Π
H
.
[Thus,
=
Π
z
H
⊆
Π
H
is
a
VCN-subgroup
of
Π
H
associated
to
z
H
∈
VCN(
H)
VCN(H)
—
cf.
Proposition
2.5,
(v).]
Then
the
following
hold:
(i)
It
holds
that
Π
z
H
=
Π
z
H
∩
Π
H
.
(ii)
Suppose
that
α
(Π
z
G
)
⊆
γ
·
Π
z
H
·
γ
−1
.
Then
there
exists
an
element
δ
∈
Π
H
such
that
α
(Π
z
G
)
⊆
δ
·
Π
z
H
·
δ
−1
.
Proof.
First,
let
us
observe
that
it
follows
immediately
from
Defini-
tion
2.3,
(ii),
together
with
the
well-known
structure
of
topological
fun-
damental
groups
of
topological
surfaces,
that
Π
z
G
and
Π
z
H
are
finitely
generated.
Thus,
it
follows
immediately
from
Theorem
2.15
that,
to
complete
the
verification
of
Corollary
2.17,
it
suffices
to
verify
that
the
following
assertion
holds:
If
Π
z
H
=
Π
H
,
then
Π
z
H
is
of
infinite
index
in
Π
H
.
To
this
end,
let
us
observe
that
since
Π
z
H
=
Π
H
,
it
follows
from
Lemma
2.16
[in
the
case
where
“G”
is
a
semi-graph
of
temperoids
of
HSD-type]
that
either
z
H
is
an
edge,
or
Node(H)
=
∅.
On
the
other
hand,
in
either
of
these
two
cases,
it
follows
immediately
from
Lemma
2.16
[in
the
case
where
“G”
is
a
semi-graph
of
anabelioids
of
PSC-type],
together
with
Proposition
2.5,
(v),
that
Π
z
H
is
of
infinite
index
in
Π
H
.
This
completes
the
proof
of
Corollary
2.17.
Corollary
2.18
(Properties
of
VCN-subgroups).
Let
G
be
a
semi-
graph
of
temperoids
of
HSD-type.
Write
Π
G
for
the
fundamental
group
of
G.
Also,
write
G
→
G
for
the
universal
covering
of
G
corresponding
to
Π
G
.
Then
the
following
hold:
[cf.
Definition
2.1,
(v)].
Write
(i)
For
i
=
1,
2,
let
v
i
∈
Vert(
G)
Π
v
i
⊆
Π
G
for
the
verticial
subgroup
of
Π
G
associated
to
v
i
[cf.
Definition
2.6,
(ii)].
Consider
the
following
three
[mutually
exclusive]
conditions
[cf.
Definition
2.1,
(v)]:
(1)
δ(
v
1
,
v
2
)
=
0.
(2)
δ(
v
1
,
v
2
)
=
1.
(3)
δ(
v
1
,
v
2
)
≥
2.
Then
we
have
equivalences
(1)
⇐⇒
(1
);
(2)
⇐⇒
(2
);
(3)
⇐⇒
(3
)
with
the
following
three
conditions:
74
YUICHIRO
HOSHI
AND
SHINICHI
MOCHIZUKI
(1
)
Π
v
1
=
Π
v
2
.
(2
)
Π
v
1
∩
Π
v
2
=
{1},
but
Π
v
1
=
Π
v
2
.
(3
)
Π
v
1
∩
Π
v
2
=
{1}.
(ii)
In
the
situation
of
(i),
suppose
that
condition
(2),
hence
also
v
1
)
∩
E(
v
2
))
=
1
condition
(2
),
holds.
Then
it
holds
that
(E(
[cf.
Definition
2.1,
(v)],
and,
moreover,
if
we
write
e
∈
E(
v
1
)
∩
E(
v
2
)
for
the
unique
element
of
E(
v
1
)
∩
E(
v
2
),
then
Π
v
1
∩
Π
v
2
=
Π
e
;
Π
e
=
Π
v
1
;
Π
e
=
Π
v
2
.
[cf.
Definition
2.1,
(v)].
Write
(iii)
For
i
=
1,
2,
let
e
i
∈
Edge(
G)
Π
e
i
⊆
Π
G
for
the
edge-like
subgroup
of
Π
G
associated
to
e
i
[cf.
Definition
2.6,
(ii)].
Then
Π
e
1
∩
Π
e
2
=
{1}
if
and
only
if
e
1
=
e
2
.
In
particular,
Π
e
1
∩Π
e
2
=
{1}
if
and
only
if
Π
e
1
=
Π
e
2
[cf.
Remark
2.6.1].
e
∈
Edge(
G).
Write
Π
v
,
Π
e
⊆
Π
G
for
the
(iv)
Let
v
∈
Vert(
G),
VCN-subgroups
of
Π
G
associated
to
v
,
e
,
respectively.
Then
Π
e
∩
Π
v
=
{1}
if
and
only
if
e
∈
E(
v
).
In
particular,
Π
e
∩
Π
v
=
{1}
if
and
only
if
Π
e
⊆
Π
v
[cf.
Remark
2.6.1].
(v)
Every
VCN-subgroup
of
Π
G
is
commensurably
terminal
in
Π
G
.
Proof.
Write
G
∧
→
G
for
the
universal
profinite
étale
covering
of
the
semi-graph
of
anabelioids
of
pro-Primes
PSC-type
G
[cf.
Proposition
2.5,
(iii),
in
the
case
where
Σ
=
Primes]
determined
by
G
→
G
and
Π
G
for
the
[profinite]
fundamental
group
of
G
determined
by
the
universal
Thus,
one
verifies
easily
that
one
obtains
a
nat-
covering
G
∧
→
G.
ural
morphism
of
[pro-]semi-graphs
of
temperoids
[cf.
Remark
2.1.1]
G
→
G
∧
that
induces
injections
Π
G
→
Π
G
[cf.
Proposition
2.5,
(iii)]
→
VCN(
G
∧
)
[cf.
[NodNon],
Definition
1.1,
(iii)]
such
and
VCN(
G)
that
→
VCN(
G
∧
)
is
compatible
with
the
re-
•
the
injection
VCN(
G)
spective
“δ’s”
[cf.
Definition
2.1,
(v);
[NodNon],
Definition
1.1,
(viii)],
and,
moreover,
the
closure
Π
z
⊆
Π
of
the
image
of
the
•
for
each
z
∈
VCN(
G),
G
VCN-subgroup
Π
z
⊆
Π
G
of
Π
G
associated
to
z
via
the
injection
Π
G
→
Π
G
coincides
with
the
VCN-subgroup
of
Π
G
[cf.
[CbTpI],
Definition
2.1,
(i)]
associated
to
the
image
of
z
via
the
injection
→
VCN(
G
∧
)
[cf.
also
Proposition
2.5,
(v)].
VCN(
G)
First,
we
verify
assertion
(i).
The
equivalence
(1)
⇔
(1
)
follows
im-
mediately
from
the
equivalence
(1)
⇔
(1
)
of
[NodNon],
Lemma
1.9,
(ii),
together
with
the
discussion
at
the
beginning
of
the
present
proof.
Next,
let
us
observe
that,
by
considering
the
edge-like
subgroup
asso-
ciated
to
an
element
of
E(
v
1
)
∩
E(
v
2
),
we
conclude
that
condition
(2)
implies
the
condition
that
Π
v
1
∩
Π
v
2
=
{1}.
Thus,
the
implication
(2)
COMBINATORIAL
ANABELIAN
TOPICS
IV
75
⇒
(2
)
follows
immediately
from
the
equivalence
(1)
⇔
(1
).
The
im-
plication
(2
)
⇒
(2)
follows
immediately
from
Corollary
2.17,
(i),
and
the
implication
(2
)
⇒
(2)
of
[NodNon],
Lemma
1.9,
(ii),
together
with
the
discussion
at
the
beginning
of
the
present
proof.
The
equivalence
(3)
⇔
(3
)
follows
immediately
from
the
equivalences
(1)
⇔
(1
)
and
(2)
⇔
(2
).
This
completes
the
proof
of
assertion
(i).
Assertion
(iii)
(respectively,
(iv))
follows
immediately
from
[NodNon],
Lemma
1.5
(respectively,
[NodNon],
Lemma
1.7),
together
with
the
discussion
at
the
beginning
of
the
present
proof.
Assertion
(v)
fol-
lows
formally
from
assertions
(i),
(iii)
[cf.
also
the
proof
of
[CmbGC],
Proposition
1.2,
(ii)].
Finally,
we
verify
assertion
(ii).
Suppose
that
condition
(2)
[in
the
statement
of
assertion
(i)],
hence
also
condition
(2
)
[in
the
statement
v
2
))
=
1
of
assertion
(i)],
holds.
Then
the
assertion
that
(E(
v
1
)
∩
E(
follows
immediately
from
the
fact
that
the
underlying
semi-graph
of
G
is
a
tree.
The
remainder
of
assertion
(ii)
follows
immediately
—
in
light
of
assertion
(iii)
—
from
Corollary
2.17,
(i),
and
[NodNon],
Lemma
1.9,
(i)
[cf.
also
Remark
2.6.1],
together
with
the
discussion
at
the
beginning
of
the
present
proof.
This
completes
the
proof
of
assertion
(ii),
hence
also
of
Corollary
2.18.
Corollary
2.19
(Graphicity
of
outer
isomorphisms).
Let
G,
H
H
for
the
semi-
be
semi-graphs
of
temperoids
of
HSD-type.
Write
G,
graphs
of
anabelioids
of
pro-Primes
PSC-type
determined
by
G,
H
[cf.
Proposition
2.5,
(iii),
in
the
case
where
Σ
=
Primes],
respectively;
Π
G
,
Π
H
for
the
respective
fundamental
groups
of
G,
H;
Π
G
,
Π
H
for
the
H.
Let
respective
[profinite]
fundamental
groups
of
G,
∼
α
:
Π
G
−→
Π
H
∼
be
an
outer
isomorphism.
Write
α
:
Π
G
→
Π
H
for
the
outer
isomor-
phism
determined
by
the
outer
isomorphism
α
and
the
natural
outer
∼
∼
G
→
H
→
isomorphisms
Π
Π
G
,
Π
Π
H
of
Proposition
2.5,
(iii).
Then
the
following
hold:
(i)
The
outer
isomorphism
α
is
group-theoretically
verticial
(respectively,
group-theoretically
cuspidal;
group-theore-
tically
nodal;
graphic)
[cf.
Definition
2.7,
(i),
(ii)]
if
and
only
if
α
is
group-theoretically
verticial
[cf.
[CmbGC],
Definition
1.4,
(iv)]
(respectively,
group-theoretically
cusp-
idal
[cf.
[CmbGC],
Definition
1.4,
(iv)];
group-theoretically
nodal
[cf.
[NodNon],
Definition
1.12];
graphic
[cf.
[CmbGC],
Definition
1.4,
(i)]).
(ii)
The
outer
isomorphism
α
is
graphic
if
and
only
if
α
is
group-
theoretically
verticial,
group-theoretically
cuspidal,
and
group-theoretically
nodal.
76
YUICHIRO
HOSHI
AND
SHINICHI
MOCHIZUKI
Proof.
Assertion
(ii)
follows
immediately,
in
light
of
Corollary
2.18,
from
a
similar
argument
to
the
argument
applied
in
the
proof
of
[CmbGC],
Proposition
1.5,
(ii).
Thus,
it
remains
to
verify
assertion
(i).
The
neces-
sity
portion
of
assertion
(i)
follows
immediately
from
Proposition
2.5,
(v).
Next,
let
us
observe
that
inclusions
of
verticial
subgroups
of
the
fundamental
group
of
a
semi-graph
of
temperoids
of
HSD-type
are
nec-
essarily
equalities
[cf.
Corollary
2.18,
(i),
(ii)];
a
similar
statement
holds
concerning
inclusions
of
edge-like
subgroups
[cf.
Corollary
2.18,
(iii)].
Thus,
the
sufficiency
portion
of
assertion
(i)
follows
immediately
—
in
light
of
assertion
(ii)
and
[CmbGC],
Proposition
1.5,
(ii)
—
from
Corollary
2.17,
(ii).
This
completes
the
proof
of
Corollary
2.19.
Corollary
2.20
(Discrete
combinatorial
cuspidalization).
Let
Σ
⊆
Primes
be
a
subset
which
is
either
equal
to
Primes
or
of
cardinal-
ity
one,
(g,
r)
a
pair
of
nonnegative
numbers
such
that
2g
−
2
+
r
>
0,
n
a
positive
integer,
and
X
a
topological
surface
of
type
(g,
r)
[i.e.,
the
complement
of
r
distinct
points
in
an
orientable
compact
topological
surface
of
genus
g].
For
each
positive
integer
i,
write
X
i
for
the
i-th
configuration
space
of
X
[i.e.,
the
topological
space
obtained
by
forming
the
complement
of
the
various
diagonals
in
the
direct
product
of
i
copies
of
X
];
Π
i
for
the
topological
fundamental
group
of
X
i
;
Π
Σ
i
for
the
pro-Σ
completion
of
Π
i
;
Π
i
for
the
profinite
completion
of
Π
i
;
Out
FC
(Π
i
)
⊆
Out
F
(Π
i
)
⊆
Out(Π
i
)
for
the
subgroups
of
the
group
Out(Π
i
)
of
outomorphisms
of
Π
i
defined
in
the
statement
of
[CmbCsp],
Corollary
5.1
[cf.
also
the
discussion
entitled
“Topological
groups”
in
[CbTpI],
§0];
F
Σ
Σ
Out
FC
(Π
Σ
i
)
⊆
Out
(Π
i
)
⊆
Out(Π
i
)
Σ
for
the
subgroups
of
the
group
Out(Π
Σ
i
)
of
outomorphisms
of
Π
i
con-
sisting
of
FC-admissible,
F-admissible
[cf.
[CmbCsp],
Definition
1.1,
(ii);
the
discussion
entitled
“Topological
groups”
in
[CbTpI],
§0]
outo-
morphisms,
respectively.
Then
the
following
hold:
(i)
The
group
Π
n
is
normally
terminal
in
Π
Σ
n
[cf.
Proposition
2.5,
(iii)].
In
particular,
the
natural
homomorphism
Out
F
(Π
n
)
−→
Out
F
(Π
Σ
n
)
is
injective.
In
the
following,
we
shall
regard
subgroups
of
Out
F
(Π
n
)
as
subgroups
of
Out
F
(Π
Σ
n
).
F
FC
(ii)
It
holds
that
Out
(Π
n
)
∩
Out
(
Π
n
)
=
Out
FC
(Π
n
).
COMBINATORIAL
ANABELIAN
TOPICS
IV
77
(iii)
Consider
the
commutative
diagram
n+1
)
Out
F
(Π
n+1
)
−−−→
Out
F
(
Π
⏐
⏐
⏐
⏐
n
)
Out
F
(Π
n
)
−−−→
Out
F
(
Π
—
where
the
horizontal
arrows
are
the
injections
of
(i),
and
the
vertical
arrows
are
the
homomorphisms
induced
by
the
pro-
jection
X
n+1
→
X
n
obtained
by
forgetting
the
(n
+
1)-st
factor.
Suppose
that
the
right-hand
vertical
arrow
of
the
diagram
is
injective
[cf.
Remark
2.20.1
below].
Then
the
commutative
diagram
of
the
above
display
is
cartesian.
In
particular,
the
left-hand
vertical
arrow
of
the
diagram
is
injective.
(iv)
The
image
of
the
left-hand
vertical
arrow
of
the
commuta-
tive
diagram
of
(iii)
[where
we
do
not
impose
the
assumption
that
the
right-hand
vertical
arrow
be
injective]
is
contained
in
Out
FC
(Π
n
)
⊆
Out
F
(Π
n
).
(v)
Consider
the
commutative
diagram
n+1
)
Out
FC
(Π
n+1
)
−−−→
Out
FC
(
Π
⏐
⏐
⏐
⏐
n
)
Out
FC
(Π
n
)
−−−→
Out
FC
(
Π
—
where
the
horizontal
arrows
are
the
injections
induced
by
the
injections
of
(i),
and
the
vertical
arrows
are
the
homo-
morphisms
induced
by
the
projection
X
n+1
→
X
n
obtained
by
forgetting
the
(n
+
1)-st
factor.
This
diagram
is
cartesian,
its
right-hand
vertical
arrow
is
injective,
and
its
left-hand
verti-
cal
arrow
is
bijective.
(vi)
Write
⎧
if
(g,
r)
=
(0,
3),
⎨
2
def
3
if
(g,
r)
=
(0,
3)
and
r
=
0,
n
FC
=
⎩
4
if
r
=
0.
Suppose
that
n
≥
n
FC
.
Then
it
holds
that
Out
FC
(Π
n
)
=
Out
F
(Π
n
);
the
left-hand
vertical
arrow
Out
F
(Π
n+1
)
−→
Out
F
(Π
n
)
of
the
commutative
diagram
of
(iii)
is
bijective.
Proof.
Let
us
first
observe
that,
to
verify
assertion
(i),
it
suffices
to
verify
that
Π
n
is
normally
terminal
in
Π
Σ
n
.
Moreover,
once
one
proves
the
desired
normal
terminality
in
the
case
where
n
=
1,
the
desired
normal
terminality
in
the
case
where
n
≥
2
follows
immediately
by
78
YUICHIRO
HOSHI
AND
SHINICHI
MOCHIZUKI
induction
[cf.
the
proof
of
[CmbCsp],
Corollary
5.1,
(i)].
Thus,
we
conclude
that,
to
verify
assertion
(i),
it
suffices
to
verify
the
normal
terminality
of
Π
1
in
Π
Σ
1
.
Next,
we
claim
that
the
following
assertion
holds:
Claim
2.20.A:
Let
F
be
a
free
nonabelian
group.
Then
F
is
normally
terminal
in
the
pro-Σ
completion
of
F
.
Indeed,
since
F
is
conjugacy
l-separable
[cf.
[Prs],
Theorem
3.2]
for
every
l
∈
Σ,
Claim
2.20.A
follows
from
a
similar
argument
to
the
argument
applied
in
the
proof
of
[André],
Lemma
3.2.1.
This
completes
the
proof
of
Claim
2.20.A.
Next,
let
us
observe
that
one
verifies
easily
that
there
exist
a
semi-
graph
of
temperoids
of
HSD-type
G
and
an
isomorphism
of
Π
1
with
the
fundamental
group
Π
G
of
G.
In
the
following,
we
shall
identify
Π
G
with
Π
1
by
means
of
such
an
isomorphism.
If
G
has
a
cusp,
then
it
follows
from
Remark
2.5.1
that
Π
1
is
a
free
nonabelian
group.
Thus,
the
desired
normal
terminality
follows
from
Claim
2.20.A.
In
the
remainder
of
the
proof
of
assertion
(i),
suppose
that
G
has
no
cusp.
In
particular,
we
may
assume
without
loss
of
generality,
by
applying
a
suitable
specialization
outer
isomorphism
[cf.
Proposition
2.10],
that
G
has
a
node.
Let
γ
∈
N
Π
Σ1
(Π
1
)
be
an
element
of
the
normalizer
of
Π
1
in
Π
Σ
and
Π
⊆
Π
v
G
a
1
verticial
subgroup
of
Π
G
.
Then,
by
applying
Corollary
2.17,
(ii)
[i.e.,
in
the
case
where
we
take
the
“(G,
H,
Π
z
H
,
Π
z
G
,
γ
)”
of
Corollary
2.17
to
be
)
and
the
“
α
”
of
Corollary
2.17
to
be
the
automorphism
(G,
G,
Π
v
,
Π
v
,
γ
],
we
conclude
immediately
[cf.
also
of
Π
G
obtained
by
conjugation
by
γ
Corollary
2.18,
(i),
(ii)]
that
we
may
assume
without
loss
of
generality,
by
multiplying
γ
by
a
suitable
element
of
Π
G
,
that
the
element
γ
∈
Π
Σ
1
normalizes
Π
v
,
hence
also
the
closure
Π
v
of
Π
v
in
Π
Σ
.
In
particular,
1
it
follows
from
Proposition
2.5,
(v);
[CmbGC],
Proposition
1.2,
(ii),
that
γ
∈
Π
v
.
On
the
other
hand,
since
G
has
a
node,
it
follows
from
Proposition
2.5,
(iv),
and
Remark
2.6.1
that
Π
v
is
a
free
nonabelian
group,
and
Π
v
may
be
identified
with
the
pro-Σ
completion
of
Π
v
.
Thus,
it
follows
from
Claim
2.20.A
that
γ
∈
Π
v
⊆
Π
G
,
as
desired.
This
completes
the
proof
of
assertion
(i).
Assertion
(ii)
follows
immediately
from
Corollary
2.19,
(i).
Next,
we
verify
assertion
(iii).
Let
us
first
observe
that
since
[we
have
assumed
that]
the
right-hand
vertical
arrow
of
the
diagram
of
assertion
(iii)
is
injective,
it
follows
immediately
from
assertion
(i)
that
all
arrows
of
the
diagram
of
assertion
(iii)
are
injective.
Let
α
∈
Out
F
(Π
n
)
be
such
that
n
)
lies
in
the
image
of
the
right-hand
vertical
the
image
of
α
in
Out
F
(
Π
arrow
of
the
diagram
of
assertion
(iii).
Then
it
follows
from
[CbTpI],
n
)
is
FC-admissible.
Theorem
A,
(ii),
that
the
image
of
α
in
Out
F
(
Π
FC
Thus,
it
follows
from
assertion
(ii)
that
α
∈
Out
(Π
n
).
In
particular,
it
follows
from
[NodNon],
Corollary
6.6,
that
there
exists
a
uniquely
COMBINATORIAL
ANABELIAN
TOPICS
IV
79
determined
element
of
Out
FC
(Π
n+1
)
whose
image
in
Out
F
(Π
n
)
coin-
cides
with
α
∈
Out
F
(Π
n
).
Thus,
since
all
arrows
of
the
diagram
of
assertion
(iii)
are
injective
[as
verified
above],
we
conclude
that
the
diagram
of
assertion
(iii)
is
cartesian.
This
completes
the
proof
of
as-
sertion
(iii).
Assertion
(iv)
follows
immediately
from
[CbTpI],
Theorem
A,
(ii),
together
with
assertion
(ii).
Assertion
(v)
follows
immediately
from
a
similar
argument
to
the
argument
applied
in
the
proof
of
asser-
tion
(iii),
together
with
the
injectivity
portion
of
[NodNon],
Theorem
B.
Assertion
(vi)
follows
immediately
from
[CbTpII],
Theorem
A,
(ii),
together
with
assertions
(i),
(ii),
(v).
This
completes
the
proof
of
Corol-
lary
2.20.
Remark
2.20.1.
It
follows
from
[CbTpII],
Theorem
A,
(i),
that
if
either
n
=
1
or
r
=
0,
then
the
right-hand
vertical
arrow
of
the
diagram
of
Corollary
2.20,
(iii),
is
injective.
Remark
2.20.2.
In
the
notation
of
Corollary
2.20,
the
bijectivity
of
the
left-hand
vertical
arrow
Out
FC
(Π
n+1
)
→
Out
FC
(Π
n
)
of
the
diagram
of
Corollary
2.20,
(v),
is
proven
in
[NodNon],
Corollary
6.6,
by
apply-
ing,
in
essence,
a
well-known
result
concerning
topological
surfaces
due
to
Dehn-Nielsen-Baer
[cf.
the
proof
of
[CmbCsp],
Corollary
5.1,
(ii)].
On
the
other
hand,
the
equivalences
of
Corollary
2.19,
(i)
[cf.
also
the
injection
of
Corollary
2.20,
(i)],
together
with
a
similar
argument
to
the
argument
applied
in
the
proof
of
the
bijectivity
portion
of
[NodNon],
Theorem
B
—
i.e.,
in
essence,
the
argument
applied
in
the
proof
of
[CmbCsp],
Corollary
3.3
—
allow
one
to
give
a
purely
algebraic
alter-
native
proof
of
this
bijectivity
result
in
the
case
where
n
≥
max{3,
n
FC
}
[cf.
Corollary
2.20,
(vi)].
Corollary
2.21
(Discrete/profinite
Dehn
multi-twists).
In
the
situation
of
Example
2.4,
(i),
write
G
X
log
for
the
semi-graph
of
an-
abelioids
of
pro-Primes
PSC-type
of
Proposition
2.5,
(iii),
in
the
case
where
we
take
“(G,
Σ)”
to
be
(G
X
log
,
Primes);
Π
G
X
log
,
Π
G
log
for
the
X
G
for
the
profi-
respective
fundamental
groups
of
G
X
log
,
G
X
log
;
Π
X
log
nite
completion
of
Π
G
X
log
[so
we
have
a
natural
outer
isomorphism
∼
G
→
Π
G
log
—
cf.
Proposition
2.5,
(iii)];
Π
X
log
X
Dehn(G
X
log
)
⊆
Out(Π
G
X
log
)
for
the
subgroup
consisting
of
the
Dehn
multi-twists
of
G
X
log
,
i.e.,
of
α
∈
Out(Π
G
X
log
)
such
that
the
following
conditions
are
satisfied:
(a)
α
is
graphic
[cf.
Definition
2.7,
(ii)]
and
induces
the
identity
automorphism
on
the
underlying
semi-graph
of
G
X
log
.
80
YUICHIRO
HOSHI
AND
SHINICHI
MOCHIZUKI
(b)
Let
Π
v
⊆
Π
G
X
log
be
a
verticial
subgroup
of
Π
G
X
log
.
Then
the
outomorphism
of
Π
v
induced
by
restricting
α
[cf.
(a);
Corol-
lary
2.18,
(v);
the
evident
discrete
analogue
of
[CbTpII],
Lemma
3.10]
is
trivial.
Then
the
following
hold:
(i)
The
composite
of
natural
outer
homomorphisms
∼
G
−→
Π
G
log
Π
G
X
log
−→
Π
X
log
X
determines
an
injection
Out(Π
G
X
log
)
→
Out(Π
G
log
).
X
(ii)
If
one
regards
subgroups
of
Out(Π
G
X
log
)
as
subgroups
of
Out(Π
G
log
)
X
by
means
of
the
injection
of
(i),
then
the
equality
Dehn(G
X
log
)
=
Dehn(
G
X
log
)
∩
Out(Π
G
X
log
)
[cf.
[CbTpI],
Definition
4.4]
holds.
(iii)
The
homomorphism
of
the
final
display
of
Example
2.4,
(i),
de-
log
(C)|
s
)
termines,
relative
to
the
natural
outer
isomorphism
π
1
(X
an
∼
→
Π
G
X
log
,
an
isomorphism
∼
log
π
1
(S
an
(C))
−→
Dehn(G
X
log
)
of
free
Z-modules
of
rank
Node(G
X
log
)
.
Moreover,
the
image
of
this
isomorphism
is
dense,
relative
to
the
profinite
topology,
in
Dehn(
G
X
log
).
Proof.
Assertion
(i)
follows
from
Corollary
2.20,
(i).
Next,
we
verify
assertion
(ii).
The
inclusion
Dehn(G
X
log
)
⊆
Dehn(
G
X
log
)
∩
Out(Π
G
X
log
)
follows
immediately
from
the
various
definitions
involved.
To
verify
the
reverse
inclusion,
let
α
∈
Dehn(
G
X
log
)
∩
Out(Π
G
X
log
).
Then
it
follows
immediately
from
Corollary
2.19,
(i),
together
with
the
definition
of
Dehn(
G
X
log
),
that
the
outomorphism
α
of
Π
G
X
log
satisfies
the
condition
(a)
in
the
statement
of
Corollary
2.21.
Moreover,
it
follows
immediately
from
Proposition
2.5,
(v),
and
Corollary
2.20,
(i),
together
with
the
definition
of
Dehn(
G
X
log
),
that
the
outomorphism
α
of
Π
G
X
log
satisfies
the
condition
(b)
in
the
statement
of
Corollary
2.21.
This
completes
the
proof
of
assertion
(ii).
Finally,
we
verify
assertion
(iii).
First,
let
us
observe
that
it
fol-
lows
immediately
from
the
various
definitions
involved
that
the
ho-
momorphism
of
the
final
display
of
Example
2.4,
(i),
factors
through
Dehn(G
X
log
)
and
has
dense
image
[i.e.,
relative
to
the
profinite
topology]
in
Dehn(
G
X
log
)
[cf.
[CbTpI],
Proposition
5.6,
(ii)].
Next,
let
us
recall
from
[CbTpI],
Theorem
4.8,
(ii),
(iv),
that
if,
for
e
∈
Node(G
X
log
)
=
def
Node(
G
X
log
),
we
write
S
e
=
Node(G
X
log
)
\
{e}
and
(G
X
log
)
∧
S
e
for
the
semi-graph
of
anabelioids
of
pro-Primes
PSC-type
of
Proposition
2.5,
COMBINATORIAL
ANABELIAN
TOPICS
IV
81
(iii),
in
the
case
where
we
take
“(G,
Σ)”
to
be
((G
X
log
)
S
e
,
Primes)
[cf.
Definition
2.9]
and
regard
Dehn((G
X
log
)
∧
S
e
)
as
a
closed
subgroup
of
Dehn(
G
X
log
)
via
the
specialization
outer
isomorphism
of
[CbTpI],
Definition
2.10
[cf.
also
Remark
2.9.1,
Proposition
2.10
of
the
present
paper],
then
we
have
an
equality
Dehn((G
X
log
)
∧
S
e
)
Dehn(
G
X
log
)
=
e∈Node(G
X
log
)
—
where
each
direct
summand
is
[noncanonically]
isomorphic
to
Z.
Here,
we
note
that
these
specialization
outer
isomorphisms
are
compat-
ible
[cf.
[CbTpI],
Proposition
5.6,
(ii),
(iii),
(iv)]
with
the
corresponding
homomorphisms
of
the
final
display
of
Example
2.4,
(i).
Thus,
in
light
of
the
density
assertion
that
has
already
been
verified,
one
verifies
im-
mediately
that,
to
complete
the
verification
of
assertion
(iii),
it
suffices
to
verify
that
the
image
of
Dehn(G
X
log
)
via
the
projection
to
any
di-
rect
summand
of
the
direct
sum
decomposition
of
the
above
display
is
contained
in
some
submodule
of
the
direct
summand
that
is
isomor-
phic
to
Z.
To
this
end,
let
us
recall
from
[CbTpI],
Theorem
4.8,
(iv),
that
such
an
image
via
a
projection
to
a
direct
summand
may
be
com-
puted
by
considering
the
homomorphism
of
the
first
display
of
[CbTpI],
Lemma
4.6,
(ii),
i.e.,
which
determines
an
isomorphism
between
the
di-
e
rect
summand
under
consideration
and
any
profinite
nodal
subgroup
Π
associated
to
the
node
e
corresponding
to
the
direct
summand.
On
the
other
hand,
it
follows
immediately
—
in
light
of
the
definition
of
this
isomorphism
—
from
Proposition
2.5,
(v);
Corollary
2.17,
(i),
that
the
image
of
Dehn(G
X
log
)
under
consideration
is
contained
in
a
suitable
discrete
nodal
subgroup
Π
e
(
∼
=
Z)
associated
to
e
[cf.
Remark
2.6.1].
This
completes
the
proof
of
assertion
(iii).
Definition
2.22.
Suppose
that
Σ
=
Primes.
Let
(g,
r)
be
a
pair
of
nonnegative
integers
such
that
2g
−
2
+
r
>
0;
n
a
positive
integer;
def
def
def
k
=
C;
S
log
=
Spec(k)
log
the
log
scheme
obtained
by
equipping
S
=
Spec(k)
with
the
log
structure
determined
by
the
fs
chart
N
→
k
that
maps
1
→
0;
X
log
=
X
1
log
a
stable
log
curve
of
type
(g,
r)
over
S
log
.
For
each
[possibly
empty]
subset
E
⊆
{1,
.
.
.
,
n},
write
X
E
log
for
the
E
-th
log
configuration
space
of
the
stable
log
curve
X
log
[cf.
the
discussion
entitled
“Curves”
in
[CbTpI],
§0],
where
we
think
of
the
factors
as
being
labeled
by
the
elements
of
E
⊆
{1,
.
.
.
,
n}
[cf.
the
discussion
at
the
beginning
of
[CbTpII],
§3,
in
the
case
where
(Σ,
k)
=
(Primes,
C)].
For
each
nonnegative
integer
n
and
each
[possibly
empty]
log
for
the
morphism
of
fs
log
subset
E
⊆
{1,
.
.
.
,
n},
write
(X
E
log
)
an
→
S
an
82
YUICHIRO
HOSHI
AND
SHINICHI
MOCHIZUKI
analytic
spaces
determined
by
the
morphism
X
E
log
→
S
log
;
(X
E
log
)
an
(C),
log
(C)
for
the
respective
topological
spaces
“X
log
”
defined
in
[KN],
S
an
(1.2),
in
the
case
where
we
take
the
“X”
of
[KN],
(1.2),
to
be
(X
E
log
)
an
,
log
log
[cf.
the
notation
established
in
Example
2.4,
(i)].
Let
s
∈
S
an
(C).
S
an
Write
def
X
E
=
(X
E
log
)
an
(C)|
s
log
for
the
fiber
of
the
natural
morphism
(X
E
log
)
an
(C)
→
S
an
(C)
at
s;
def
Π
disc
=
π
1
(X
E
)
E
for
the
discrete
topological
fundamental
group
of
X
E
;
def
def
def
=
Π
disc
X
n
=
X
{1,...,n}
;
X
=
X
1
;
Π
disc
n
{1,...,n}
.
Thus,
for
sets
E
⊆
E
⊆
{1,
.
.
.
,
n},
we
have
a
projection
p
an
E/E
:
X
E
→
X
E
obtained
by
forgetting
the
factors
that
belong
to
E
\
E
.
For
sets
E
⊆
E
⊆
{1,
.
.
.
,
n}
and
nonnegative
integers
m
≤
n,
write
disc
disc
disc
p
Π
E/E
:
Π
E
Π
E
for
some
fixed
surjection
[that
belongs
to
the
collection
of
surjections
that
constitutes
the
outer
surjection]
induced
by
p
an
E/E
;
def
disc
Π
disc
Π
disc
E/E
=
Ker(p
E/E
)
⊆
Π
E
def
an
p
an
n/m
=
p
{1,...,n}/{1,...,m}
:
X
n
−→
X
m
;
disc
def
disc
Π
disc
Π
disc
p
Π
m
;
n/m
=
p
{1,...,n}/{1,...,m}
:
Π
n
def
disc
disc
Π
disc
n/m
=
Π
{1,...,n}/{1,...,m}
⊆
Π
n
.
disc
”
for
the
profinite
completion
of
“Π
disc
”.
Finally,
we
shall
write
“
Π
(−)
(−)
Thus,
we
have
a
natural
outer
isomorphism
∼
disc
−→
Π
Π
E
E
—
where
Π
E
is
as
in
the
discussion
at
the
beginning
of
[CbTpII],
§3.
def
def
log
;
Π
n
=
Π
{1,...,n}
.
In
the
following,
we
shall
also
write
X
n
log
=
X
{1,...,n}
Definition
2.23.
In
the
notation
of
Definition
2.22,
let
i
∈
E
⊆
{1,
.
.
.
,
n};
x
∈
X
n
(C)
a
C-valued
geometric
point
of
the
underlying
scheme
X
n
of
X
n
log
.
(i)
We
shall
write
G
disc
for
the
semi-graph
of
temperoids
of
HSD-type
associated
to
X
log
[cf.
Example
2.4,
(ii)];
disc
G
i∈E,x
COMBINATORIAL
ANABELIAN
TOPICS
IV
83
for
the
semi-graph
of
temperoids
of
HSD-type
associated
to
the
geometric
fiber
[cf.
Example
2.4,
(ii);
Remark
2.4.1]
of
the
log
log
log
log
projection
p
log
E/(E\{i})
:
X
E
→
X
E\{i}
over
x
E\{i}
→
X
E\{i}
[cf.
[CbTpII],
Definition
3.1,
(i)];
Π
G
disc
,
Π
G
i∈E,x
disc
disc
for
the
respective
fundamental
groups
of
G
disc
,
G
i∈E,x
[cf.
Propo-
sition
2.5,
(i)];
G
disc
Π
i∈E,x
for
the
profinite
completion
of
Π
G
i∈E,x
disc
.
Thus,
it
follows
from
the
discussion
of
Remark
2.5.2
that
we
have
a
natural
graphic
[cf.
[CmbGC],
Definition
1.4,
(i)]
outer
isomorphism
∼
G
disc
−→
Π
Π
G
i∈E,x
i∈E,x
—
where
G
i∈E,x
is
the
semi-graph
of
anabelioids
of
pro-Primes
PSC-type
of
[CbTpII],
Definition
3.1,
(iii)
—
and
hence
a
nat-
ural
isomorphism
of
semi-graphs
of
anabelioids
∼
G
disc
−→
G
i∈E,x
i∈E,x
disc
—
where
we
write
G
i∈E,x
for
the
semi-graph
of
anabelioids
of
pro-Primes
PSC-type
of
Proposition
2.5,
(iii),
in
the
case
disc
,
Primes).
Moreover,
it
where
we
take
“(G,
Σ)”
to
be
(G
i∈E,x
follows
immediately
from
the
discussion
of
Example
2.4
that
we
have
a
natural
Π
disc
E
-orbit
[i.e.,
relative
to
composition
with
automorphisms
induced
by
conjugation
by
elements
of
Π
disc
E
]
of
isomorphisms
∼
disc
(Π
disc
disc
.
E
⊇)
Π
E/(E\{i})
−→
Π
G
i∈E,x
One
verifies
immediately
from
the
various
definitions
involved
that
the
diagram
∼
disc
disc
Π
E/(E\{i})
−−−→
Π
G
i∈E,x
⏐
⏐
⏐
⏐
∼
Π
E/(E\{i})
−−−→
Π
G
i∈E,x
disc
-
—
where
the
upper
horizontal
arrow
is
an
element
of
the
Π
E
disc
orbit
of
isomorphisms
induced
by
the
Π
E
-orbit
of
isomor-
phisms
of
the
above
discussion;
the
lower
horizontal
arrow
is
an
element
of
the
Π
E
-orbit
of
isomorphisms
of
[CbTpII],
Def-
inition
3.1,
(iii);
the
left-hand
vertical
arrow
is
the
isomor-
phism
obtained
by
forming
the
restriction
of
an
isomorphism
∼
disc
→
Π
Π
E
that
belongs
to
the
outer
isomorphism
of
the
fi-
E
nal
display
of
Definition
2.22;
the
right-hand
vertical
arrow
is
an
isomorphism
that
belongs
to
the
outer
isomorphism
of
the
84
YUICHIRO
HOSHI
AND
SHINICHI
MOCHIZUKI
above
discussion
—
commutes
up
to
composition
with
auto-
morphisms
induced
by
conjugation
by
elements
of
Π
E
.
disc
(ii)
We
shall
say
that
a
vertex
v
∈
Vert(G
i∈E,x
)
is
a(n)
[E-]tripod
of
X
n
if
v
is
of
type
(0,
3)
[cf.
Definition
2.6,
(iii)].
Thus,
one
disc
)
is
a(n)
[E-]tripod
if
and
verifies
easily
that
v
∈
Vert(G
i∈E,x
only
if
the
corresponding
vertex
of
G
i∈E,x
via
the
graphic
outer
∼
G
disc
→
isomorphism
Π
Π
G
i∈E,x
of
(i)
is
a(n)
[E-]tripod
of
X
n
log
i∈E,x
[cf.
[CbTpII],
Definition
3.1,
(v)].
We
shall
refer
to
a
verticial
subgroup
of
Π
G
i∈E,x
associated
to
a(n)
[E-]tripod
of
X
n
as
a(n)
disc
disc
[E-]tripod
of
Π
n
.
(iii)
Let
P
be
a
property
of
[E-]tripods
of
Π
n
[cf.
[CbTpII],
Defi-
nition
3.3,
(i)]
or
X
n
log
[e.g.,
the
property
of
being
strict
—
cf.
[CbTpII],
Definition
3.3,
(iii);
the
property
of
arising
from
an
edge
—
cf.
[CbTpII],
Definition
3.7,
(i);
the
property
of
being
central
—
cf.
[CbTpII],
Definition
3.7,
(ii)].
Then
we
shall
say
that
a(n)
[E-]tripod
of
Π
disc
or
X
n
[cf.
(ii)]
satisfies
P
if
the
n
corresponding
[E-]tripod
of
Π
n
or
X
n
log
satisfies
P.
(iv)
Let
T
⊆
Π
disc
be
an
E-tripod
of
Π
disc
[cf.
(ii)].
Then
one
may
n
E
define
the
subgroups
Out
C
(T
),
Out
C
(T
)
cusp
,
Out
C
(T
)
Δ
,
Out
C
(T
)
Δ+
⊆
Out(T
)
of
Out(T
)
in
an
entirely
analogous
fashion
to
the
definition
of
the
closed
subgroups
“Out
C
(T
)”,
“Out
C
(T
)
cusp
”,
“Out
C
(T
)
Δ
”,
“Out
C
(T
)
Δ+
”
of
“Out(T
)”
given
in
[CbTpII],
Definition
3.4,
(i).
We
leave
the
routine
details
to
the
reader.
Theorem
2.24
(Outomorphisms
preserving
tripods).
In
the
no-
tation
of
Definition
2.22,
let
E
⊆
{1,
.
.
.
,
n}
be
a
subset
and
T
⊆
Π
disc
E
an
E-tripod
of
Π
disc
[cf.
Definition
2.23,
(ii)].
Let
us
write
n
F
disc
Out
F
(Π
disc
n
)[T
]
⊆
Out
(Π
n
)
for
the
subgroup
of
Out
F
(Π
disc
n
)
[cf.
the
notational
conventions
intro-
duced
in
the
statement
of
Corollary
2.20]
consisting
of
α
∈
Out
F
(Π
disc
n
)
disc
such
that
the
outomorphism
of
Π
E
determined
by
α
preserves
the
disc
Π
disc
E
-conjugacy
class
of
T
⊆
Π
E
;
def
F
FC
FC
disc
disc
disc
Out
FC
(Π
disc
n
)[T
]
=
Out
(Π
n
)[T
]
∩
Out
(Π
n
)
⊆
Out
(Π
n
)
[cf.
the
notational
conventions
introduced
in
the
statement
of
Corol-
def
def
def
C
disc
)
=
Out
FC
(Π
disc
);
lary
2.20];
Π
=
Π
1
;
Π
disc
=
Π
disc
1
;
Out
(Π
def
Out
C
(Π)
=
Out
FC
(Π).
Then
the
following
hold:
(i)
Write
T
for
the
profinite
completion
of
T
.
Then
the
natural
homomorphism
Out(T
)
−→
Out(
T
)
COMBINATORIAL
ANABELIAN
TOPICS
IV
85
is
injective.
If,
moreover,
one
regards
subgroups
of
Out(T
)
as
subgroups
of
Out(
T
)
via
this
injection,
then
it
holds
that
Out
C
(T
)
=
Out
C
(
T
)
∩
Out(T
),
Out
C
(T
)
cusp
=
Out
C
(
T
)
cusp
∩
Out(T
),
Out
C
(T
)
Δ
=
Out
C
(
T
)
Δ
∩
Out(T
),
Out
C
(T
)
Δ+
=
Out
C
(
T
)
Δ+
∩
Out(T
)
[cf.
Definition
2.23,
(iv);
[CbTpII],
Definition
3.4,
(i)].
(ii)
It
holds
that
Out
C
(T
)
cusp
=
Out
C
(T
)
Δ
=
Out
C
(T
)
Δ+
∼
=
Z/2Z,
Out
C
(T
)
∼
=
Z/2Z
×
S
3
—
where
we
write
S
3
for
the
symmetric
group
on
3
letters.
(iii)
The
commensurator
and
centralizer
of
T
∈
Π
disc
satisfy
the
E
equality
C
Π
disc
(T
)
=
T
×
Z
Π
disc
(T
).
E
E
Thus,
by
applying
the
evident
discrete
analogue
of
[CbTpII],
Lemma
3.10,
to
outomorphisms
of
Π
disc
determined
by
ele-
E
F
disc
ments
of
Out
(Π
n
)[T
],
one
obtains
a
natural
homomorphism
T
T
:
Out
F
(Π
disc
n
)[T
]
−→
Out(T
).
(iv)
Suppose
that
n
≥
3,
and
that
T
is
central
[cf.
Definition
2.23,
(iii)].
Then
it
holds
that
F
disc
Out
F
(Π
disc
n
)
=
Out
(Π
n
)[T
].
Moreover,
the
homomorphism
F
disc
T
T
:
Out
F
(Π
disc
n
)
=
Out
(Π
n
)[T
]
−→
Out(T
)
of
(iii)
determines
a
surjection
C
Δ+
∼
(
=
Z/2Z).
Out
FC
(Π
disc
n
)
Out
(T
)
We
shall
refer
to
this
homomorphism
as
the
tripod
homo-
morphism
associated
to
Π
disc
n
.
(v)
The
profinite
completion
T
determines
an
E-tripod
of
Π
n
,
which,
by
abuse
of
notation,
we
denote
by
T
.
Now
suppose
that
T
is
E-strict
[cf.
Definition
2.23,
(iii)].
Then
it
holds
that
F
F
disc
Out
F
(Π
disc
n
)[T
]
=
Out
(Π
n
)[
T
]
∩
Out
(Π
n
)
[cf.
[CbTpII],
Theorem
3.16].
86
YUICHIRO
HOSHI
AND
SHINICHI
MOCHIZUKI
(vi)
Suppose
that
the
semi-graph
of
anabelioids
of
pro-Primes
PSC-
type
G
associated
to
X
log
[cf.
[CbTpII],
Definition
3.1,
(ii)]
is
totally
degenerate
[cf.
[CbTpI],
Definition
2.3,
(iv)].
Re-
call
that
G
may
be
naturally
identified
with
the
semi-graph
of
anabelioids
of
pro-Primes
PSC-type
determined
by
G
disc
[cf.
Proposition
2.5,
(iii);
the
discussion
of
Definition
2.23,
(i)].
Then
one
has
an
equality
Aut(G
disc
)
−
=
Aut(G)
∩
Out
C
(Π
disc
)
−
(⊆
Out
C
(Π))
—
where
the
superscript
“
−
’s”
denote
the
closure
in
the
profi-
nite
topology
—
of
subgroups
of
Out
C
(Π)
[cf.
Corollary
2.20,
(i)].
Proof.
First,
we
verify
assertion
(i).
The
injectivity
portion
of
asser-
tion
(i)
follows
from
Corollary
2.20,
(i).
The
first
equality
follows
from
Corollary
2.20,
(ii).
Thus,
the
second
and
third
equalities
follow
imme-
diately
from
the
various
definitions
involved;
the
fourth
equality
follows
from
Corollary
2.20,
(v).
This
completes
the
proof
of
assertion
(i).
Next,
we
verify
assertion
(ii).
The
inclusions
Out
C
(T
)
Δ+
⊆
Out
C
(T
)
Δ
⊆
Out
C
(T
)
cusp
follow
from
assertion
(i),
together
with
[CbTpII],
Lemma
3.5.
The
inclusion
Out
C
(T
)
cusp
⊆
Out
C
(T
)
Δ+
and
the
assertion
that
Out
C
(T
)
cusp
∼
=
Z/2Z
follow
immediately
from
[CmbCsp],
Corollary
5.3,
(i),
together
with
a
classical
result
of
Nielsen
[cf.
[CmbCsp],
Remark
5.3.1].
This
completes
the
proof
of
the
first
line
of
the
display
of
as-
sertion
(ii).
Now
since
Out
C
(T
)
Δ
=
Out
C
(T
)
cusp
,
by
considering
the
action
of
Out
C
(T
)
on
the
set
of
the
T
-conjugacy
classes
of
cuspidal
inertia
subgroups
of
T
,
we
obtain
an
exact
sequence
1
−→
Out
C
(T
)
Δ
−→
Out
C
(T
)
−→
S
3
−→
1.
By
considering
outomorphisms
of
T
arising
from
automorphisms
of
analytic
spaces,
one
obtains
a
section
of
this
sequence;
moreover,
it
follows
from
the
definition
of
Out
C
(T
)
Δ
that
this
section
determines
∼
an
isomorphism
Out
C
(T
)
Δ
×
S
3
→
Out
C
(T
).
This
completes
the
proof
of
assertion
(ii).
Next,
we
verify
assertion
(iii).
Recall
that
every
finite
index
subgroup
of
T
is
normally
terminal
in
its
profinite
completion
[cf.
Corollary
2.20,
(i)]
and
center-free
[cf.
Remark
2.6.1].
Thus,
assertion
(iii)
follows
immediately
from
[CbTpII],
Theorem
3.16,
(i).
This
completes
the
proof
of
assertion
(iii).
Next,
we
verify
assertion
(iv).
First,
let
us
observe
that
it
fol-
lows
immediately
from
the
definition
of
the
notion
of
a
central
tri-
pod
[cf.
Definition
2.23,
(iii);
[CbTpII],
Definition
3.7,
(ii)]
that
we
may
assume
without
loss
of
generality
that
n
=
3.
To
verify
the
equality
of
the
first
display
of
assertion
(iv),
we
mimick
the
argu-
ment
in
the
profinite
case
given
in
the
proof
of
[CmbCsp],
Corollary
∈
Aut(Π
disc
1.10,
(i):
Let
α
∈
Out
F
(Π
disc
n
),
α
n
)
a
lifting
of
α.
Write
COMBINATORIAL
ANABELIAN
TOPICS
IV
87
α
2
∈
Aut(Π
disc
.
Now
observe
that
2
)
for
the
automorphism
induced
by
α
F
disc
since
α
∈
Out
(Π
n
),
it
follows
immediately
from
Corollary
2.20,
(iv),
2
preserves
that
α
2
determines
an
element
of
Out
FC
(Π
disc
2
),
hence
that
α
the
Π
disc
-conjugacy
class
of
inertia
groups
associated
to
the
diagonal
2
[cf.
Definition
2.22;
the
discussion
of
cusp
of
any
of
the
fibers
of
p
an
2/1
[CmbCsp],
Remark
1.1.5].
Thus,
by
replacing
α
by
the
composite
of
α
with
a
suitable
inner
automorphism,
we
may
assume
without
loss
of
generality
that
α
2
preserves
the
inertia
group
associated
to
some
F
disc
diagonal
cusp
of
a
fiber
of
p
an
2/1
.
Now
the
fact
that
α
∈
Out
(Π
n
)[T
]
follows
immediately
from
Corollary
2.17,
(ii);
[CbTpII],
Theorem
1.9,
(ii)
[cf.
the
application
of
[CmbCsp],
Proposition
1.3,
(iv),
in
the
proof
of
[CmbCsp],
Corollary
1.10,
(i)].
The
assertion
that
the
restriction
to
F
disc
Out
FC
(Π
disc
n
)
of
the
homomorphism
Out
(Π
n
)
→
Out(T
)
of
assertion
(iii)
factors
through
Out
C
(T
)
Δ+
⊆
Out(T
)
follows
immediately
from
from
assertions
(i)
and
(ii),
together
with
[CbTpII],
Theorem
3.16,
(v).
The
assertion
that
the
resulting
homomorphism
is
surjective
fol-
lows
immediately
from
the
fact
that
the
[unique]
nontrivial
element
of
Out
C
(T
)
Δ+
is
the
outomorphism
induced
by
complex
conjugation
[cf.
[CmbCsp],
Remark
5.3.1],
together
with
the
[easily
verified]
fact
that
the
pointed
stable
curve
over
C
corresponding
to
the
given
stable
log
curve
X
log
may
be
assumed,
without
loss
of
generality
—
i.e.,
by
apply-
ing
a
suitable
specialization
isomorphism
[cf.
the
discussion
preceding
[CmbCsp],
Definition
2.1,
as
well
as
[CbTpI],
Remark
5.6.1]
and
ob-
serving
that
such
specialization
isomorphisms
are
compatible
with
the
various
discrete
fundamental
groups
involved
[cf.
Remarks
2.9.1
and
2.10.1]
—
to
be
defined
over
R.
This
completes
the
proof
of
assertion
(iv).
Next,
we
verify
assertion
(v).
It
follows
immediately
from
the
clas-
sification
of
E-strict
tripods
given
in
[CbTpII],
Lemma
3.8,
(ii),
that
we
may
assume
without
loss
of
generality
that
E
=
n
≤
3.
When
n
=
3,
assertion
(v)
follows
formally
from
assertion
(iv).
When
n
=
1,
assertion
(v)
follows
immediately
from
Corollary
2.17,
(ii).
Thus,
it
remains
to
consider
the
case
where
n
=
2,
i.e.,
where
the
tripod
T
arises
from
an
edge.
In
this
case,
assertion
(v)
follows
from
a
similar
argument
to
the
argument
applied
in
the
proof
of
assertion
(iv).
That
is
to
say,
let
α
∈
Out
F
(Π
disc
∈
Aut(Π
disc
2
),
α
2
)
a
lifting
of
α.
Write
disc
α
1
∈
Aut(Π
1
)
for
the
automorphism
induced
by
α
;
β
1
∈
Aut(Π
1
),
.
Then
we
must
β
∈
Aut(Π
2
)
for
the
automorphisms
determined
by
α
F
disc
verify
that
α
∈
Out
(Π
2
)[T
]
under
the
assumption
that
β
deter-
mines
an
element
β
∈
Out
F
(Π
2
)[
T
].
Now
observe
that
it
follows
im-
mediately
from
the
computation
of
the
centralizer
given
in
[CbTpII],
Lemma
3.11,
(vii),
that
β
1
preserves
the
Π
1
-conjugacy
class
of
edge-like
subgroups
of
Π
1
determined
by
the
edge
that
gives
rise
to
the
tripod
T
.
Thus,
we
conclude
from
Corollary
2.17,
(ii),
that,
by
replacing
α
88
YUICHIRO
HOSHI
AND
SHINICHI
MOCHIZUKI
by
the
composite
of
α
with
a
suitable
inner
automorphism,
we
may
corre-
assume
that
α
1
preserves
a
specific
edge-like
subgroup
of
Π
disc
1
sponding
to
the
edge
that
gives
rise
to
the
tripod
T
.
Note
that
this
assumption
implies,
in
light
of
the
commensurably
terminality
of
edge-
like
subgroups
[cf.
[CmbGC],
Proposition
1.2,
(ii)],
that
β
preserves
the
Π
2/1
-conjugacy
class
of
the
tripod
T
.
In
particular,
we
conclude,
as
in
the
proof
of
assertion
(iv),
i.e.,
by
applying
Corollary
2.17,
(ii),
that
α
∈
Out
F
(Π
disc
2
)[T
],
as
desired.
This
completes
the
proof
of
assertion
(v).
Finally,
we
verify
assertion
(vi).
First,
let
us
observe
that
it
follows
immediately
from
Corollary
2.20,
(v),
that
both
sides
of
the
equality
FC
C
−
in
question
are
⊆
Out
FC
(Π
disc
3
)
⊆
Out
(Π
3
)
(⊆
Out
(Π)).
Also,
we
observe
that,
by
considering
the
case
where
X
log
is
defined
over
R
[cf.
the
proof
of
assertion
(iv)],
it
follows
immediately
that
both
sides
of
the
equality
in
question
surject,
via
the
tripod
homomorphism
of
assertion
(iv),
onto
the
finite
group
of
order
two
that
appears
as
the
image
of
this
tripod
homomorphism
[cf.
also
the
fact
that
the
topological
group
Out(
T
)
is
profinite,
hence,
in
particular,
Hausdorff].
In
particular,
to
complete
the
proof
of
assertion
(v),
it
suffices
to
verify
that
the
evident
inclusion
Aut(G
disc
)
−
∩Out
FC
(Π
3
)
geo
⊆
Aut(G)
∩
Out
C
(Π
disc
)
−
∩
Out
FC
(Π
3
)
geo
—
where
we
write
Out
FC
(Π
3
)
geo
⊆
Out
FC
(Π
3
)
for
the
kernel
of
the
tripod
homomorphism
on
Out
FC
(Π
3
)
[cf.
[CbTpII],
Definition
3.19]
—
of
subgroups
of
Out
C
(Π)
is,
in
fact,
an
equality.
On
the
other
hand,
since
Dehn(G)
is
a
normal
open
subgroup
of
both
Aut(G
disc
)
−
∩
Out
FC
(Π
3
)
geo
and
Aut(G)
∩
Out
C
(Π
disc
)
−
∩
Out
FC
(Π
3
)
geo
[cf.
Corol-
lary
2.21,
(iii);
[CbTpI],
Theorem
4.8,
(i);
the
commutative
diagram
of
[CbTpII],
Corollary
3.27,
(ii)],
and
Aut(G
disc
)
−
∩
Out
FC
(Π
3
)
geo
clearly
surjects
onto
the
finite
group
of
automorphisms
of
the
underlying
semi-
graph
of
G
disc
,
the
desired
equality
follows
immediately
from
[CbTpII],
Corollary
3.27,
(ii).
This
completes
the
proof
of
assertion
(vi).
Remark
2.24.1.
It
is
not
clear
to
the
authors
at
the
time
of
writing
whether
or
not
one
can
remove
the
strictness
assumption
imposed
in
Theorem
2.24,
(v).
Indeed,
from
the
point
of
view
of
induction
on
n,
the
essential
difficulty
in
removing
this
assumption
may
already
be
seen
in
the
case
of
a
non-E-strict
tripod
when
E
=
n
=
2.
From
another
point
of
view,
this
difficulty
may
be
thought
of
as
arising
from
the
lack
of
an
analogue
for
discrete
topological
fundamental
groups
of
n-th
configuration
spaces,
when
n
≥
2,
of
Corollary
2.17.
COMBINATORIAL
ANABELIAN
TOPICS
IV
89
Remark
2.24.2.
(i)
In
the
notation
of
Theorem
2.24,
let
us
observe
that
it
follows
from
Corollary
2.19,
(i),
that
we
have
an
equality
Aut(G
disc
)
=
Aut(G)
∩
Out
C
(Π
disc
)
(⊆
Out
C
(Π))
of
subgroups
of
Out
C
(Π)
[cf.
Corollary
2.20,
(i)].
On
the
other
hand,
it
is
by
no
means
clear
whether
or
not
the
evident
inclu-
sion
Aut(G
disc
)
−
⊆
Aut(G)
∩
Out
C
(Π
disc
)
−
(⊆
Out
C
(Π))
(∗)
—
where
the
superscript
“
−
’s”
denote
the
closure
in
the
profi-
nite
topology
—
is
an
equality
in
general.
On
the
other
hand,
when
X
log
is
totally
degenerate,
this
equality
is
the
content
of
Theorem
2.24,
(vi).
(ii)
We
continue
to
use
the
notation
of
(i).
Write
M
Q
for
the
moduli
stack
of
hyperbolic
curves
of
type
(g,
r)
over
Q
and
C
Q
→
M
Q
for
the
tautological
hyperbolic
curve
over
M
Q
.
Thus,
def
for
appropriate
choices
of
basepoints,
if
we
write
Π
C
=
π
1
(C
Q
),
def
Π
M
=
π
1
(M
Q
)
for
the
respective
étale
fundamental
groups,
then
we
obtain
an
exact
sequence
of
profinite
groups
1
−→
Δ
C/M
−→
Π
C
−→
Π
M
−→
1
—
where
Δ
C/M
is
defined
so
as
to
render
the
sequence
exact
—
as
well
as
a
natural
outer
representation
ρ
M
:
Π
M
−→
Out
C
(Π)
—
where,
by
choosing
appropriate
basepoints,
we
identify
Π
with
Δ
C/M
—
and
a
natural
outer
surjection
Π
M
G
Q
onto
the
absolute
Galois
group
G
Q
of
Q
[cf.
the
discussion
of
[CbTpII],
Remark
3.19.1].
Write
G
R
⊆
G
Q
for
the
decomposi-
tion
group
[which
is
well-defined
up
to
G
Q
-conjugation]
of
the
unique
archimedean
prime
of
Q.
In
the
spirit
of
[Bgg1],
[Bgg2],
[Bgg3],
let
us
write
def
Γ
=
Out
C
(Π
disc
)
(⊆
Out
C
(Π));
def
Γ̌
=
ρ
M
(Π
M
×
G
Q
G
R
)
[cf.
Corollary
2.20,
(i)].
Thus,
for
appropriate
choices
of
base-
points,
Γ̌
is
equal
to
the
closure
of
Γ
in
Out
C
(Π).
If
σ
is
a
sim-
plex
of
the
complex
of
profinite
curves
L(Π)
studied
in
[Bgg1],
[Bgg2],
[Bgg3],
that
arises
from
Π
disc
,
then
the
stabilizer
in
Γ
of
σ
is
denoted
Γ
σ
,
while
the
stabilizer
in
Γ̌
of
the
image
of
σ
in
the
profinite
curve
complex
corresponding
to
Γ̌
is
denoted
Γ̌
σ
.
Then
[Bgg3],
Theorem
4.2
[cf.
also
[Bgg1],
Proposition
6.5],
asserts
that
90
YUICHIRO
HOSHI
AND
SHINICHI
MOCHIZUKI
The
natural
inclusion
Γ
−
σ
⊆
Γ̌
σ
is,
in
fact,
an
equality.
Translated
into
the
language
of
the
present
paper,
this
asser-
tion
corresponds
precisely
to
the
assertion
that
the
inclusion
(∗)
considered
in
(i)
is,
in
fact,
an
equality.
In
particular,
The-
orem
2.24,
(vi),
corresponds,
essentially,
to
a
special
case
[i.e.,
the
totally
degenerate
case]
of
[Bgg3],
Theorem
4.2.
At
a
more
concrete
level,
when
Node(G)
=
1,
and
σ
arises
from
a
single
simple
closed
curve
that
corresponds
to
the
unique
node
e
of
G,
this
assertion
corresponds
precisely
to
the
assertion
that
the
profinite
stabilizer
in
Γ̌
of
the
Π-conjugacy
class
of
nodal
subgroups
of
Π
determined
by
e
coin-
cides
with
the
closure
in
Γ̌
of
the
discrete
stabilizer
in
Γ
of
the
Π
disc
-conjugacy
class
of
nodal
subgroups
of
Π
disc
determined
by
e
—
cf.
Theorem
3.3,
Remark
3.3.1,
Corollary
3.4
in
§3
below.
As
discussed
in
(i),
this
sort
of
assertion
is
highly
nontriv-
ial.
That
is
to
say,
this
sort
of
coincidence
between
a
profinite
stabilizer
and
the
closure
of
a
corresponding
discrete
stabilizer
is,
in
fact,
false
in
general,
as
the
example
given
in
(iv)
be-
low
demonstrates.
In
particular,
this
sort
of
coincidence
is
by
no
means
a
consequence
of
superficial
“general
nonsense”-type
considerations,
but
rather,
when
true
[cf.,
e.g.,
the
case
treated
in
Theorem
2.24,
(vi)],
a
consequence
of
deep
properties
of
the
specific
groups
and
specific
spaces
[on
which
these
groups
act]
under
consideration.
(iii)
In
closing,
we
observe
that
many
of
the
results
derived
in
[Bgg3]
as
a
consequence
of
the
assertion
discussed
in
(ii)
were,
in
fact,
already
obtained
in
earlier
papers
by
the
authors.
Indeed,
the
faithfulness
asserted
in
[Bgg3],
Theorem
7.7
—
i.e.,
the
injectivity
of
the
restriction
of
ρ
M
to
a
section
G
F
→
Π
M
arising
from
a
hyperbolic
curve
of
type
(g,
r)
defined
over
a
number
field
F
—
is
a
special
case
of
[NodNon],
Theorem
C.
On
the
other
hand,
in
[CbTpI],
Theorem
D,
a
computation
is
given
of
the
centralizer
in
Out
C
(Π)
of
an
open
subgroup
of
Γ̌.
Thus,
the
computation
of
centers
given
in
[Bgg3],
Corollary
6.2,
amounts
to
a
special
case
of
[CbTpI],
Theorem
D.
Finally,
[Bgg3],
Corollary
7.6
—
which
may
be
regarded
as
the
assertion
that
the
inverse
image
via
ρ
M
of
the
centralizer
of
Γ̌
in
Out
C
(Π)
maps
trivially
to
G
Q
—
amounts
to
a
concatenation
of
•
the
computation
of
the
centralizer
given
in
[CbTpI],
The-
orem
D,
with
•
the
fact,
stated
in
[NodNon],
Corollary
6.4,
that
ρ
−1
M
(Γ̌)
maps
trivially
to
G
Q
.
COMBINATORIAL
ANABELIAN
TOPICS
IV
91
(iv)
Let
n
≥
3
be
an
integer.
Consider
the
natural
conjuga-
tion
action
of
the
special
linear
group
SL
n
(Z)
with
coefficients
∈
Z
on
the
module
M
n
(Z)
of
n
by
n
matrices
with
coefficients
∈
Z.
Write
A
∈
M
n
(Z)
for
the
diagonal
matrix
whose
entries
are
given
by
the
integers
1,
.
.
.
,
n.
Then
one
verifies
immedi-
ately
that
the
stabilizer
SL
n
(Z)
A
of
A,
relative
to
the
conjugacy
action
of
SL
n
(Z),
is
equal
to
the
subgroup
of
diagonal
matrices
of
SL
n
(Z),
hence
isomorphic
to
the
finite
group
given
by
a
product
of
n
−
1
copies
of
the
finite
group
of
order
two
{±1}.
On
the
other
hand,
if
one
considers
with
coefficients
the
action
of
the
special
linear
group
SL
n
(
Z)
∈
Z
on
the
module
M
n
(
Z)
of
n
by
n
matrices
with
coefficients
then
one
verifies
immediately
that
the
stabilizer
∈
Z,
A
SL
n
(
Z)
is
equal
to
of
A,
relative
to
the
conjugacy
action
of
SL
n
(
Z),
hence
isomorphic
the
subgroup
of
diagonal
matrices
of
SL
n
(
Z),
×
,
a
group
of
uncountable
to
a
product
of
n
−
1
copies
of
Z
cardinality.
That
is
to
say,
A
is
much
larger
The
profinite
stabilizer
SL
n
(
Z)
than
the
profinite
completion
of
the
discrete
stabi-
lizer
SL
n
(Z)
A
.
Here,
we
recall
that
since,
as
is
well-known,
the
congruence
subgroup
problem
has
been
resolved
in
the
affirmative,
in
the
may
be
identified
case
of
n
≥
3,
the
topological
group
SL
n
(
Z)
with
the
profinite
completion
of
the
group
SL
n
(Z).
A
simi-
lar
example
may
be
given
in
the
case
of
the
symplectic
group
Sp
2n
(Z).
Corollary
2.25
(Characterization
of
the
archimedean
local
Ga-
lois
groups
in
the
global
Galois
image
associated
to
a
hyper-
bolic
curve).
Let
F
be
a
number
field
[i.e.,
a
finite
extension
of
the
field
of
rational
numbers];
p
an
archimedean
prime
of
F
;
F
p
an
algebraic
closure
of
the
p-adic
completion
F
p
of
F
[so
F
p
is
isomor-
phic
to
C];
F
⊆
F
p
the
algebraic
closure
of
F
in
F
p
;
X
F
log
a
smooth
def
def
log
curve
over
F
.
Write
G
p
=
Gal(F
p
/F
p
)
⊆
G
F
=
Gal(F
/F
);
def
def
def
=
X
F
log
×
F
F
p
;
X
F
log
=
X
F
log
×
F
F
p
;
X
F
log
=
X
F
log
×
F
F
;
X
F
log
p
p
π
1
(X
F
log
)
92
YUICHIRO
HOSHI
AND
SHINICHI
MOCHIZUKI
for
the
log
fundamental
group
of
X
F
log
;
π
1
disc
(X
F
log
)
p
for
the
[discrete]
topological
fundamental
group
of
the
analytic
space
associated
to
the
interior
of
the
log
scheme
X
F
log
;
p
π
1
disc
(X
F
log
)
∧
p
for
the
profinite
completion
of
π
1
disc
(X
F
log
);
p
ρ
X
log
:
G
F
−→
Out(π
1
(X
F
log
))
F
for
the
natural
outer
Galois
action
associated
to
X
F
log
;
ρ
disc
:
G
p
−→
Out(π
1
disc
(X
F
log
))
X
log
,p
p
F
.
Thus,
we
have
for
the
natural
outer
Galois
action
associated
to
X
F
log
p
a
natural
outer
isomorphism
∼
π
1
disc
(X
F
log
)
∧
−→
π
1
(X
F
log
),
p
which
determines
a
natural
injection
Out(π
1
disc
(X
F
log
))
→
Out(π
1
(X
F
log
))
p
[cf.
Corollary
2.20,
(i)].
Then
the
following
hold:
(i)
We
have
a
natural
commutative
diagram
ρ
disc
log
X
,p
ρ
log
F
G
p
−−−
−→
Out(π
1
disc
(X
F
log
))
p
⏐
⏐
⏐
⏐
X
F
G
F
−−−
→
Out(π
1
(X
F
log
))
—
where
the
vertical
arrows
are
the
natural
inclusions,
and
all
arrows
are
injective.
(ii)
The
diagram
of
(i)
is
cartesian,
i.e.,
if
we
regard
the
various
groups
involved
as
subgroups
of
Out(π
1
(X
F
log
)),
then
we
have
an
equality
G
p
=
G
F
∩
Out(π
1
disc
(X
F
log
)).
p
Proof.
Assertion
(i)
follows
immediately
from
the
injectivity
of
the
lower
horizontal
arrow
ρ
X
log
[cf.
[NodNon],
Theorem
C],
together
with
F
the
various
definitions
involved.
Finally,
we
verify
assertion
(ii).
Write
(X
F
)
log
3
for
the
3-rd
log
con-
log
figuration
space
of
X
F
.
Then
it
follows
immediately
from
[NodNon],
Theorem
B,
that
the
group
Out
FC
(π
1
((X
F
)
log
3
))
of
FC-admissible
outo-
log
morphisms
of
the
log
fundamental
group
π
1
((X
F
)
log
3
)
of
(X
F
)
3
may
be
COMBINATORIAL
ANABELIAN
TOPICS
IV
93
regarded
as
a
closed
subgroup
of
Out(π
1
(X
F
log
)).
Moreover,
it
follows
immediately
from
the
various
definitions
involved
that
the
respective
images
Im(ρ
X
log
),
Im(ρ
disc
)
of
the
natural
outer
Galois
actions
ρ
X
log
,
X
log
,p
F
F
F
associated
to
X
F
log
,
X
F
log
are
contained
in
this
closed
subgroup
ρ
disc
log
p
,p
X
F
log
Out
FC
(π
1
((X
F
)
log
3
))
⊆
Out(π
1
(X
F
)).
Thus,
to
verify
assertion
(ii),
one
verifies
immediately
from
Corollary
2.20,
(v),
that
it
suffices
to
verify
the
equality
)
=
Im(ρ
X
log
)
∩
Out(π
1
disc
((X
F
p
)
log
Im(ρ
disc
3
))
X
log
,p
F
F
def
log
log
disc
—
where
we
write
(X
F
p
)
log
3
=
(X
F
)
3
×
F
F
p
and
π
1
((X
F
p
)
3
)
for
the
[discrete]
topological
fundamental
group
of
the
analytic
space
associated
to
the
interior
of
the
log
scheme
(X
F
p
)
log
3
.
On
the
other
hand,
since
the
“ρ
X
log
”
that
occurs
in
the
case
where
we
take
“X
F
log
”
to
be
the
F
smooth
log
curve
associated
to
P
1
F
\
{0,
1,
∞}
is
injective
[cf.
assertion
(i)],
this
equality
follows
immediately
—
by
considering
the
images
of
the
subgroups
)
⊆
Im(ρ
X
log
)
∩
Out(π
1
disc
((X
F
p
)
log
Im(ρ
disc
3
))
X
log
,p
F
F
of
Out(π
1
disc
((X
F
p
)
log
3
))
via
the
[manifestly
compatible!]
tripod
homo-
morphisms
associated
to
π
1
disc
((X
F
p
)
log
3
)
[cf.
Theorem
2.24,
(iv)]
and
log
π
1
((X
F
)
3
)
[cf.
[CbTpII],
Theorem
3.16,
(i),
(v)]
—
from
[André],
The-
orem
3.3.1.
This
completes
the
proof
of
assertion
(ii),
hence
also
of
Corollary
2.25.
Remark
2.25.1.
Corollary
2.25
is
a
generalization
of
[André],
Theo-
rem
3.3.2
[cf.
also
the
footnote
of
[André]
following
[André],
Theorem
3.3.2].
Although
the
proof
given
here
of
Corollary
2.25
is
by
no
means
the
first
proof
of
this
result
[cf.
the
discussion
of
this
footnote
of
[André]
following
[André],
Theorem
3.3.2;
[NodNon],
Corollary
6.4],
it
is
of
in-
terest
to
note
that
this
result
may
also
be
derived
in
the
context
of
the
theory
of
the
present
paper,
i.e.,
via
an
argument
that
parallels
the
proof
given
in
[CbTpIII]
of
[CbTpIII],
Theorem
B,
in
the
p-adic
case
[for
which
no
alternative
proofs
are
known!].
94
YUICHIRO
HOSHI
AND
SHINICHI
MOCHIZUKI
3.
Canonical
liftings
of
cycles
In
the
present
§3,
we
discuss
certain
canonical
liftings
of
cycles
[cf.
Theorems
3.10,
3.14
below].
These
canonical
liftings
are
constructed
in
a
fashion
illustrated
in
Figure
1.
This
approach
to
constructing
such
canonical
liftings
was
motivated
[cf.
Remark
3.10.1
below]
by
the
ar-
guments
of
[Bgg2],
where
these
canonical
liftings
were
applied,
in
the
context
of
the
congruence
subgroup
problem
for
hyperelliptic
modular
groups,
to
derive
certain
injectivity
results
[cf.
[Bgg2],
§2],
which
may
be
regarded
as
special
cases
of
[NodNon],
Theorem
B.
Unfortunately,
however,
the
authors
of
the
present
paper
were
unable
to
follow
in
detail
these
arguments
of
[Bgg2],
which
appear
to
be
based
to
a
substantial
extent
on
geometric
intuition
concerning
the
geometry
of
topological
surfaces.
Although,
in
the
development
of
the
present
series
of
pa-
pers
on
combinatorial
anabelian
geometry,
the
authors
were
motivated
by
similar
geometric
intuition,
the
proofs
of
the
results
given
in
the
present
series
of
papers
proceed
by
means
of
purely
combinatorial
and
algebraic
arguments
concerning
combinatorial
[e.g.,
graphs]
and
group-
theoretic
[e.g.,
profinite
fundamental
groups]
data
that
arises
from
a
pointed
stable
curve.
From
the
point
of
view
of
arithmetic
geome-
try,
the
geometric
intuition
which
underlies
the
topological
arguments
given
in
[Bgg2]
involving
objects
such
as
topological
Dehn
twists
is
of
an
essentially
archimedean
nature,
hence,
in
particular,
is
fundamentally
incompatible,
at
least
from
the
point
of
view
of
establishing
a
rigorous
mathematical
formulation,
with
the
highly
nonarchimedean
properties
of
profinite
fundamental
groups,
as
studied
in
the
present
series
of
papers
—
cf.
the
discussion
of
[SemiAn],
Remark
1.5.1.
It
was
this
state
of
affairs
that
motivated
the
authors
to
give,
in
the
present
§3,
a
formulation
of
the
constructions
of
[Bgg2],
§2,
in
terms
of
the
purely
combinatorial
and
algebraic
techniques
developed
in
the
present
series
of
papers.
In
the
present
§3,
let
(g,
r)
be
a
pair
of
nonnegative
integers
such
that
2g
−
2
+
r
>
0;
n
a
positive
integer;
Σ
a
set
of
prime
numbers
which
is
either
equal
to
the
entire
set
of
prime
numbers
or
of
cardinality
one;
k
def
an
algebraically
closed
field
of
characteristic
∈
Σ;
S
log
=
Spec(k)
log
the
def
log
scheme
obtained
by
equipping
S
=
Spec(k)
with
the
log
structure
determined
by
the
fs
chart
N
→
k
that
maps
1
→
0;
X
log
=
X
1
log
a
stable
log
curve
of
type
(g,
r)
over
S
log
.
For
each
[possibly
empty]
subset
E
⊆
{1,
.
.
.
,
n},
write
X
E
log
for
the
E
-th
log
configuration
space
of
the
stable
log
curve
X
log
[cf.
the
discussion
entitled
“Curves”
in
[CbTpI],
§0],
where
we
think
of
the
factors
as
being
labeled
by
the
elements
of
E
⊆
{1,
.
.
.
,
n};
Π
E
COMBINATORIAL
ANABELIAN
TOPICS
IV
95
for
the
maximal
pro-Σ
quotient
of
the
kernel
of
the
natural
surjection
π
1
(X
E
log
)
π
1
(S
log
);
log
log
Π
p
log
E/E
:
X
E
→
X
E
,
p
E/E
:
Π
E
Π
E
,
def
def
def
log
log
Π
E/E
=
Ker(p
Π
E/E
)
⊆
Π
E
,
X
n
=
X
{1,...,n}
,
Π
n
=
Π
{1,...,n}
,
def
log
log
log
p
log
n/m
=
p
{1,...,n}/{1,...,m}
:
X
n
−→
X
m
,
def
Π
p
Π
n/m
=
p
{1,...,n}/{1,...,m}
:
Π
n
Π
m
,
def
Π
n/m
=
Π
{1,...,n}/{1,...,m}
⊆
Π
n
,
G,
G,
Π
G
,
G
i∈E,x
,
Π
G
i∈E,x
for
the
objects
defined
in
the
discussion
at
the
beginning
of
[CbTpII],
§3;
[CbTpII],
Definition
3.1.
In
addition,
we
suppose
that
we
have
been
given
a
pair
of
nonnegative
integers
(
Y
g,
Y
r)
such
that
2
Y
g
−
2
+
Y
r
>
0
and
a
stable
log
curve
Y
log
=
Y
1
log
of
type
(
Y
g,
Y
r)
over
S
log
.
We
shall
use
similar
notation
log
log
Y
Π
p
E/E
:
Y
Π
E
Y
Π
E
,
Y
E
log
,
Y
Π
E
,
Y
p
log
E/E
:
Y
E
→
Y
E
,
Y
def
def
def
log
Y
log
Π
E/E
=
Ker(
Y
p
Π
=
Y
{1,...,n}
,
Y
Π
n
=
Y
Π
{1,...,n}
,
E/E
)
⊆
Π
E
,
Y
n
Y
log
def
Y
log
p
n/m
=
p
{1,...,n}/{1,...,m}
:
Y
n
log
−→
Y
m
log
,
def
Y
Π
Y
Y
p
n/m
=
Y
p
Π
{1,...,n}/{1,...,m}
:
Π
n
Π
m
,
def
Π
n/m
=
Y
Π
{1,...,n}/{1,...,m}
⊆
Y
Π
n
,
Y
G,
Y
G,
Π
Y
G
,
Y
G
i∈E,y
,
Π
Y
G
i∈E,y
Y
for
objects
associated
to
the
stable
log
curve
Y
log
=
Y
1
log
to
the
nota-
tion
introduced
above
for
X
log
[cf.
the
discussion
at
the
beginning
of
[CbTpII],
§3;
[CbTpII],
Definition
3.1].
Lemma
3.1
(Graphicity
in
the
case
of
a
single
node).
In
the
notation
of
the
discussion
at
the
beginning
of
the
present
§3,
suppose
that
Node(G)
=
Node(
Y
G)
=
1.
Write
e
∈
Node(G)
(respectively,
Y
e
∈
Node(
Y
G))
for
the
unique
node
of
G
(respectively,
Y
G).
Let
Π
e
⊆
Π
1
(respectively,
∼
∼
Π
Y
e
⊆
Y
Π
1
)
be
a
nodal
subgroup
of
Π
1
→
Π
G
(respectively,
Y
Π
1
→
Π
Y
G
)
associated
to
e
∈
Node(G)
(respectively,
Y
e
∈
Node(
Y
G));
e
2
∈
X
2
(k)
(respectively,
Y
e
2
∈
Y
2
(k))
a
k-valued
point
of
the
underlying
scheme
X
2
(respectively,
Y
2
)
of
the
log
scheme
X
2
log
(respectively,
Y
2
log
)
that
Y
log
lies,
relative
to
p
log
2/1
(respectively,
p
2/1
),
over
the
k-valued
point
of
X
(respectively,
Y
)
determined
by
the
node
e
∈
Node(G)
(respectively,
Y
e
∈
Node(
Y
G)).
Thus,
we
obtain
an
outer
isomorphism
∼
Π
2/1
−→
Π
G
2∈{1,2},e
2
∼
(respectively,
Y
Π
2/1
→
Π
Y
G
2∈{1,2},Y
e
)
2
96
YUICHIRO
HOSHI
AND
SHINICHI
MOCHIZUKI
[cf.
[CbTpII],
Definition
3.1,
(iii)]
that
may
be
characterized,
up
to
composition
with
elements
of
Aut
|grph|
(G
2∈{1,2},e
2
)
⊆
Out(Π
G
2∈{1,2},e
2
)
(respectively,
Aut
|grph|
(
Y
G
2∈{1,2},
Y
e
2
)
⊆
Out(Π
Y
G
2∈{1,2},Y
e
))
[cf.
[CbTpI],
2
Definition
2.6,
(i);
[CbTpII],
Remark
4.1.2],
as
the
group-theoretically
cuspidal
[cf.
[CmbGC],
Definition
1.4,
(iv)]
outer
isomorphism
such
that
the
semi-graph
of
anabelioids
structure
on
G
2∈{1,2},e
2
(respectively,
Y
G
2∈{1,2},
Y
e
2
)
is
the
semi-graph
of
anabelioids
structure
determined
[cf.
[NodNon],
Theorem
A]
by
the
resulting
composite
outer
representa-
tion
∼
Π
e
→
Π
1
→
Out(Π
2/1
)
→
Out(Π
G
2∈{1,2},e
2
)
∼
(respectively,
Π
Y
e
→
Y
Π
1
→
Out(
Y
Π
2/1
)
→
Out(Π
Y
G
2∈{1,2},Y
e
))
2
—
where
the
second
arrow
is
the
outer
action
determined
by
the
exact
sequence
1
→
Π
2/1
→
Π
2
→
Π
1
→
1
(respectively,
1
→
Y
Π
2/1
→
Y
Π
2
→
Y
Π
1
→
1)
—
in
a
fashion
compatible
with
the
restriction
Π
2/1
Π
{2}
Y
Π
(respectively,
Y
Π
2/1
Y
Π
{2}
)
of
p
Π
{1,2}/{2}
(respectively,
p
{1,2}/{2}
)
to
Π
2/1
⊆
Π
2
(respectively,
Y
Π
2/1
⊆
Y
Π
2
)
and
the
given
outer
isomor-
∼
∼
∼
∼
phisms
Π
{2}
→
Π
1
→
Π
G
(respectively,
Y
Π
{2}
→
Y
Π
1
→
Y
Π
G
).
Let
(respectively,
Y
v
∈
Vert(
Y
G
2∈{1,2},
Y
e
2
))
v
∈
Vert(G
2∈{1,2},e
2
)
be
the
{1,
2}-tripod
[cf.
[CbTpII],
Definition
3.1,
(v)]
that
arises
from
e
∈
Node(G)
(respectively,
Y
e
∈
Node(
Y
G))
[cf.
[CbTpII],
Defini-
∼
tion
3.7,
(i)];
Π
v
⊆
Π
G
2∈{1,2},e
2
←
Π
2/1
(respectively,
Π
Y
v
⊆
Π
Y
G
2∈{1,2},Y
e
2
∼
←
Y
Π
2/1
)
a
{1,
2}-tripod
in
Π
2
(respectively,
Y
Π
2
)
associated
to
the
tri-
pod
v
(respectively,
Y
v)
[cf.
[CbTpII],
Definition
3.3,
(i)];
∼
α
:
Π
G
−→
Π
Y
G
an
outer
isomorphism
of
profinite
groups.
Suppose
that
the
following
conditions
are
satisfied:
(a)
The
outer
isomorphism
α
is
group-theoretically
nodal
[cf.
[NodNon],
Definition
1.12],
i.e.,
determines
a
bijection
of
the
set
of
Π
G
-conjugates
of
Π
e
⊆
Π
G
and
the
set
of
Π
Y
G
-conjugates
of
Π
Y
e
⊆
Π
Y
G
.
(b)
The
outer
isomorphism
α
is
2-cuspidalizable
[cf.
[CbTpII],
Definition
3.20],
i.e.,
the
outer
isomorphism
∼
α
∼
∼
Π
1
−→
Π
G
−→
Π
Y
G
←−
Y
Π
1
arises
from
a
[uniquely
determined,
up
to
permutation
of
the
2
factors
—
cf.
[NodNon],
Theorem
B]
PFC-admissible
[cf.
∼
[CbTpI],
Definition
1.4,
(iii)]
outer
isomorphism
Π
2
→
Y
Π
2
.
[In
particular,
the
outer
isomorphism
α
is
group-theoretically
cuspidal.]
Then
the
following
hold:
COMBINATORIAL
ANABELIAN
TOPICS
IV
97
∼
(i)
There
exists
a
PFC-admissible
isomorphism
α
2
:
Π
2
→
Y
Π
2
that
lifts
α
such
that
the
composite
∼
∼
∼
Π
G
2∈{1,2},e
2
←−
Π
2/1
−→
Y
Π
2/1
−→
Π
Y
G
2∈{1,2},Y
e
2
—
where
the
second
arrow
is
the
restriction
of
α
2
—
is
graphic
[cf.
[CmbGC],
Definition
1.4,
(i)].
∼
(ii)
The
outer
isomorphism
α
2
:
Π
2
→
Y
Π
2
determined
by
the
iso-
morphism
α
2
of
(i)
induces
a
bijection
between
the
set
of
Π
2
-
conjugates
of
Π
v
⊆
Π
2
and
the
set
of
Y
Π
2
-conjugates
of
Π
Y
v
⊆
Y
Π
2
.
Moreover,
if
we
think
of
Π
v
,
Π
Y
v
as
the
respective
[pro-Σ]
fundamental
groups
of
G
2∈{1,2},e
2
|
v
,
Y
G
2∈{1,2},
Y
e
2
|
Y
v
[cf.
[CbTpI],
Definition
2.1,
(iii);
[CbTpI],
Remark
2.1.1],
then
the
induced
∼
outer
isomorphism
Π
v
→
Π
Y
v
[cf.
[CbTpII],
Theorem
3.16,
(i)]
is
group-theoretically
cuspidal.
(iii)
The
outer
isomorphism
α
is
graphic.
Proof.
In
light
of
conditions
(a)
and
(b),
assertion
(i)
follows
immedi-
ately
from
[NodNon],
Theorem
A
[cf.
also
our
assumption
that
Node(G)
=
Node(
Y
G)
=
1,
which
implies
that
the
outer
representation
Π
e
→
Out(Π
G
2∈{1,2},e
2
)
(respectively,
Π
Y
e
→
Out(Π
Y
G
2∈{1,2},Y
e
))
is
nodally
non-
2
degenerate!].
Next,
let
us
observe
that
the
Π
G
2∈{1,2},e
2
-
(respectively,
Π
Y
G
2∈{1,2},Y
e
-)
conjugacy
class
of
Π
v
⊆
Π
G
2∈{1,2},e
2
(respectively,
Π
Y
v
⊆
2
Π
Y
G
2∈{1,2},Y
e
)
may
be
characterized
as
the
unique
Π
G
2∈{1,2},e
2
-
(respectively,
2
Π
Y
G
2∈{1,2},Y
e
-)
conjugacy
class
of
verticial
subgroups
that
fails
to
map
2
injectively
via
the
surjection
Π
2/1
Π
{2}
(respectively,
Y
Π
2/1
Y
Π
{2}
).
Now
assertion
(ii)
follows
immediately
from
assertion
(i).
Assertion
(iii)
follows
immediately
—
in
light
of
[CmbCsp],
Proposition
1.2,
(iii)
—
from
assertions
(i),
(ii),
together
with
the
various
definitions
involved.
This
completes
the
proof
of
Lemma
3.1.
Before
proceeding,
we
pause
to
observe
that
Lemma
3.1
may
be
applied
to
obtain
an
alternative
proof
of
a
slightly
weaker
version
of
Theorem
3.3
below,
as
follows.
Proposition
3.2
(Graphicity
of
group-theoretically
nodal
2-cus-
pidalizable
outer
isomorphisms).
In
the
notation
of
the
discussion
at
the
beginning
of
the
present
§3,
let
∼
α
:
Π
G
−→
Π
Y
G
be
an
outer
isomorphism
of
profinite
groups.
Suppose
that
the
following
conditions
are
satisfied:
(a)
The
outer
isomorphism
α
is
group-theoretically
nodal
[cf.
[NodNon],
Definition
1.12].
98
YUICHIRO
HOSHI
AND
SHINICHI
MOCHIZUKI
(b)
The
outer
isomorphism
α
is
2-cuspidalizable
[cf.
[CbTpII],
Definition
3.20],
i.e.,
the
outer
isomorphism
α
∼
∼
∼
Π
1
−→
Π
G
−→
Π
Y
G
←−
Y
Π
1
arises
from
a
[uniquely
determined,
up
to
permutation
of
the
2
factors
—
cf.
[NodNon],
Theorem
B]
PFC-admissible
[cf.
∼
[CbTpI],
Definition
1.4,
(iii)]
outer
isomorphism
Π
2
→
Y
Π
2
.
[In
particular,
the
outer
isomorphism
α
is
group-theoretically
cuspidal
—
cf.
[CmbGC],
Definition
1.4,
(iv).]
Then
the
outer
isomorphism
α
is
graphic
[cf.
[CmbGC],
Definition
1.4,
(i)].
Proof.
Let
us
first
observe
that
it
follows
from
condition
(a),
together
with
[CmbGC],
Proposition
1.2,
(i),
that
α
determines
a
bijection
∼
Node(G)
→
Node(
Y
G),
so
Node(G)
=
Node(
Y
G)
.
We
verify
Propo-
sition
3.2
by
induction
on
Node(G)
=
Node(
Y
G)
.
If
Node(G)
=
Node(
Y
G)
=
∅,
then
Proposition
3.2
is
immediate.
Thus,
we
may
assume
without
loss
of
generality
that
Node(G),
Node(
Y
G)
=
∅.
Let
e
∈
Node(G).
Write
Y
e
∈
Node(
Y
G)
for
the
node
of
Y
G
that
corresponds,
via
α,
to
e.
Write
G
{e}
(respectively,
Y
G
{
Y
e}
)
for
the
generization
of
G
(respectively,
Y
G)
with
respect
to
{e}
⊆
Node(G)
(respectively,
{
Y
e}
⊆
Node(
Y
G))
[cf.
[CbTpI],
Definition
2.8];
β
for
the
composite
outer
isomorphism
Φ
G
{e}
∼
α
∼
Π
G
{e}
−→
Π
G
−→
Π
Y
G
Φ
−1
Y
G
{
Y
e}
∼
−→
Π
Y
G
{
Y
e}
[cf.
[CbTpI],
Definition
2.10];
v
0
∈
Vert(G
{e}
)
(respectively,
Y
v
0
∈
Vert(
Y
G
{
Y
e}
))
for
the
[uniquely
determined]
vertex
of
the
generiza-
tion
G
{e}
(respectively,
Y
G
{
Y
e}
)
that
does
not
arise
from
a
vertex
of
Vert(G)
(respectively,
Vert(
Y
G)).
Let
Π
v
0
⊆
Π
G
{e}
(respectively,
Π
Y
v
0
⊆
Π
Y
G
{
Y
e}
)
be
a
verticial
subgroup
associated
to
v
0
∈
Vert(G
{e}
)
(respectively,
Y
v
0
∈
Vert(
Y
G
{
Y
e}
));
Π
e
⊆
Π
v
0
(respectively,
Π
Y
e
⊆
Π
Y
v
0
)
a
subgroup
that
maps
to
a
nodal
subgroup
associated
to
e
in
Π
G
(respectively,
to
Y
e
in
Π
Y
G
).
Thus,
it
follows
immediately
from
[NodNon],
Lemma
1.9,
(i),
(ii)
[cf.
also
[NodNon],
Lemma
1.5;
con-
dition
(2)
of
[CbTpI],
Proposition
2.9,
(i)],
that
Π
v
0
(respectively,
Π
Y
v
0
)
may
be
characterized
as
the
unique
verticial
subgroup
of
Π
G
{e}
(respectively,
Π
Y
G
{
Y
e}
)
that
contains
Π
e
(respectively,
Π
Y
e
).
Next,
let
us
observe
that,
by
applying
the
induction
hypothesis
to
β,
we
conclude
that
β
is
graphic.
Thus,
it
follows
immediately
—
in
light
of
[CmbGC],
Proposition
1.5,
(ii)
—
from
the
definition
of
the
gener-
izations
under
consideration
[cf.
condition
(3)
of
[CbTpI],
Proposition
2.9,
(i)]
that,
to
complete
the
verification
of
Proposition
3.2,
it
suffices
to
verify
that
the
following
assertion
holds:
COMBINATORIAL
ANABELIAN
TOPICS
IV
99
Claim
3.2.A:
Let
H
⊆
Π
v
0
⊆
Π
G
{e}
be
a
closed
sub-
group
of
Π
v
0
whose
image
in
Π
G
is
a
verticial
subgroup.
Then
the
image
of
H
via
the
composite
β
∼
Π
G
{e}
−→
Π
Y
G
{
Y
e}
Φ
Y
G
{
Y
e}
∼
−→
Π
Y
G
is
a
verticial
subgroup.
To
verify
Claim
3.2.A,
let
us
observe
that
since
β
is
graphic,
it
fol-
lows
immediately
from
the
above
characterization
of
Π
v
0
,
Π
Y
v
0
that
β
maps
Π
v
0
bijectively
onto
a
Π
Y
G
{
Y
e}
-conjugate
of
Π
Y
v
0
.
Thus,
it
follows
immediately
from
condition
(b),
together
with
the
evident
iso-
morphism
[i.e.,
as
opposed
to
outomorphism
—
cf.
[CbTpII],
Remark
4.14.1]
version
of
[CbTpII],
Lemma
4.8,
(i),
(ii),
that,
in
the
notation
∼
of
[CbTpII],
Definition
4.3,
the
outer
isomorphism
Π
2
→
Y
Π
2
of
con-
∼
dition
(b)
induces
compatible
outer
isomorphisms
(Π
v
0
)
2
→
(Π
Y
v
0
)
2
,
∼
Π
v
0
→
Π
Y
v
0
.
In
particular,
by
applying
Lemma
3.1,
(iii),
to
these
outer
isomorphisms,
one
concludes
that
Claim
3.2.A
holds,
as
desired.
This
completes
the
proof
of
Proposition
3.2.
Theorem
3.3
(Graphicity
of
profinite
outer
isomorphisms).
Let
Σ
0
be
a
nonempty
set
of
prime
numbers;
H,
J
semi-graphs
of
anabe-
lioids
of
pro-Σ
0
PSC-type;
Π
H
,
Π
J
the
[pro-Σ
0
]
fundamental
groups
of
H,
J
,
respectively;
∼
α
:
Π
H
−→
Π
J
an
outer
isomorphism
of
profinite
groups.
Then
the
following
condi-
tions
are
equivalent:
(i)
The
outer
isomorphism
α
is
graphic
[cf.
[CmbGC],
Definition
1.4,
(i)].
(ii)
The
outer
isomorphism
α
is
group-theoretically
verticial
and
group-theoretically
cuspidal
[cf.
[CmbGC],
Definition
1.4,
(iv)].
(iii)
The
outer
isomorphism
α
is
group-theoretically
nodal
[cf.
[NodNon],
Definition
1.12]
and
group-theoretically
cuspi-
dal.
Proof.
The
implication
(i)
⇒
(ii)
(respectively,
(ii)
⇒
(iii))
follows
from
the
various
definitions
involved
(respectively,
[NodNon],
Lemma
1.9,
(i)).
Thus,
it
suffices
to
verify
the
implication
(iii)
⇒
(i).
Suppose
that
condition
(iii)
holds.
Then,
to
verify
the
graphicity
of
α,
it
follows
from
[CmbGC],
Theorem
1.6,
(ii),
that
it
suffices
to
verify
that
α
is
graphically
filtration-preserving
[cf.
[CmbGC],
Definition
1.4,
(iii)].
In
particular,
by
replacing
Π
H
,
Π
J
by
suitable
open
subgroups
of
Π
H
,
Π
J
,
100
YUICHIRO
HOSHI
AND
SHINICHI
MOCHIZUKI
it
suffices
to
verify
that
α
determines
isomorphisms
∼
∼
Π
ab-edge
−→
Π
ab-edge
,
Π
ab-vert
−→
Π
ab-vert
H
J
H
J
—
where
we
write
“Π
ab-edge
”,
“Π
ab-vert
”
for
the
closed
subgroups
of
(−)
(−)
ab
the
abelianization
“Π
(−)
”
of
“Π
(−)
”
topologically
generated
by
the
im-
ages
of
the
edge-like,
verticial
subgroups
of
“Π
(−)
”.
Here,
we
may
as-
sume
without
loss
of
generality
that
H
and
J
are
sturdy,
hence
admit
compactifications
[cf.
[CmbGC],
Remarks
1.1.5,
1.1.6].
Now
the
asser-
”
follows
immediately
from
condition
(iii).
On
tion
concerning
“Π
ab-edge
(−)
the
other
hand,
the
assertion
concerning
“Π
ab-vert
”
follows
immediately
(−)
from
the
duality
discussed
in
[CmbGC],
Proposition
1.3,
applied
to
the
compactifications
of
H,
J
,
together
with
condition
(iii).
This
completes
the
proof
of
Theorem
3.3.
Remark
3.3.1.
Here,
we
observe
that
results
such
as
[Bgg3],
Corol-
lary
6.1;
[Bgg3],
Corollary
6.4,
(ii);
[Bgg3],
Theorem
6.6,
amount,
when
translated
into
the
language
of
the
present
paper,
to
a
special
case
of
the
result
obtained
by
concatenating
the
equivalence
(i)
⇔
(iii)
of
The-
orem
3.3,
with
the
computation
of
the
normalizer
given
in
[CbTpI],
Theorem
5.14,
(iii)
[i.e.,
in
essence,
[CmbGC],
Corollary
2.7,
(iii),
(iv)].
Moreover,
the
proof
given
above
of
this
equivalence
(i)
⇔
(iii)
of
Theo-
rem
3.3
is,
essentially,
a
restatement
of
various
results
from
the
theory
of
[CmbGC].
That
is
to
say,
although
the
statements
of
these
results
that
occur
in
the
present
series
of
papers
and
in
[Bgg3]
are
formulated
and
arranged
in
a
somewhat
different
way,
the
essential
mathematical
content
that
underlies
these
results
is,
in
fact,
entirely
identical;
more-
over,
this
state
of
affairs
is
by
no
means
a
coincidence.
Indeed,
this
mathematical
content
is
given
in
[CmbGC]
as
[CmbGC],
Proposition
1.3;
[CmbGC],
Proposition
2.6.
In
[Bgg3],
this
mathematical
content
is
given
as
[Bgg3],
Lemma
5.11
[and
the
surrounding
discussion],
which,
in
fact,
was
related
to
the
author
of
[Bgg3]
by
the
senior
author
of
the
present
paper
in
the
context
of
an
explanation
of
the
theory
of
[CmbGC].
Corollary
3.4
(Graphicity
of
discrete
outer
isomorphisms).
Let
H,
J
be
semi-graphs
of
temperoids
of
HSD-type
[cf.
Definition
2.3,
(iii)];
Π
H
,
Π
J
the
fundamental
groups
of
H,
J
,
respectively
[cf.
Propo-
sition
2.5,
(i)];
∼
α
:
Π
H
−→
Π
J
an
outer
isomorphism.
Then
the
following
conditions
are
equivalent:
(i)
The
outer
isomorphism
α
is
graphic
[cf.
Definition
2.7,
(ii)].
COMBINATORIAL
ANABELIAN
TOPICS
IV
101
(ii)
The
outer
isomorphism
α
is
group-theoretically
verticial
and
group-theoretically
cuspidal
[cf.
Definition
2.7,
(i)].
(iii)
The
outer
isomorphism
α
is
group-theoretically
nodal
and
group-theoretically
cuspidal
[cf.
Definition
2.7,
(i)].
Proof.
This
follows
immediately
from
Theorem
3.3,
together
with
Corol-
lary
2.19,
(i).
∼
Definition
3.5.
Let
(
Y
G,
S
⊆
Node(
Y
G),
φ
:
Y
G
S
→
G)
be
a
degener-
ation
structure
on
G
[cf.
[CbTpII],
Definition
3.23,
(i)]
and
e
∈
S.
(i)
We
shall
say
that
a
closed
subgroup
J
⊆
Π
1
of
Π
1
is
a
cycle-
∼
subgroup
of
Π
1
[with
respect
to
(
Y
G,
S
⊆
Node(
Y
G),
φ
:
Y
G
S
→
G),
associated
to
e
∈
S]
if
J
is
contained
in
the
Π
1
-conjugacy
class
of
closed
subgroups
of
Π
1
obtained
by
forming
the
image
of
a
nodal
subgroup
of
Π
Y
G
associated
to
e
via
the
composite
of
outer
isomorphisms
Φ
−1
Y
G
S
∼
∼
∼
Π
Y
G
−→
Π
Y
G
S
−→
Π
G
−→
Π
1
—
where
the
first
arrow
is
the
inverse
of
the
specialization
outer
isomorphism
Φ
Y
G
S
[cf.
[CbTpI],
Definition
2.10],
the
second
∼
arrow
is
the
graphic
outer
isomorphism
Π
Y
G
S
→
Π
G
induced
by
φ,
and
the
third
arrow
is
the
natural
outer
isomorphism
∼
Π
G
→
Π
1
of
[CbTpII],
Definition
3.1,
(ii)
[cf.
the
left-hand
portion
of
Figure
1].
(ii)
Let
n
be
a
positive
integer.
Then
we
shall
say
that
a
cycle-
subgroup
of
Π
1
is
n-cuspidalizable
if
it
is
a
cycle-subgroup
of
Π
1
with
respect
to
some
n-cuspidalizable
degeneration
structure
on
G
[cf.
[CbTpII],
Definition
3.23,
(v)].
Remark
3.5.1.
Let
J
⊆
Π
1
be
a
cycle-subgroup
of
Π
1
with
respect
∼
to
a
degeneration
structure
(
Y
G,
S
⊆
Node(
Y
G),
φ
:
Y
G
S
→
G),
associ-
ated
to
a
node
e
∈
S.
Then
it
follows
immediately
from
[CmbGC],
Proposition
1.2,
(i),
that
the
node
e
of
Y
G
is
uniquely
determined
by
the
subgroup
J
⊆
Π
1
and
the
degeneration
structure
(
Y
G,
S
⊆
∼
Node(
Y
G),
φ
:
Y
G
S
→
G).
Definition
3.6.
Let
J
⊆
Π
1
be
a
2-cuspidalizable
cycle-subgroup
of
Π
1
[cf.
Definition
3.5,
(i),
(ii)].
(i)
It
follows
immediately
from
the
various
definitions
involved
that
we
have
data
as
follows:
102
YUICHIRO
HOSHI
AND
SHINICHI
MOCHIZUKI
(a)
a
2-cuspidalizable
degeneration
structure
(
Y
G,
S
⊆
Node(
Y
G),
∼
φ
:
Y
G
S
→
G)
on
G
[cf.
[CbTpII],
Definition
3.23,
(i),
(v)],
∼
(b)
an
isomorphism
Y
Π
1
→
Π
1
that
is
compatible
with
the
composite
of
the
display
of
Definition
3.5,
(i)
[cf.
also
[CbTpII],
Definition
3.1,
(ii)],
in
the
case
where
we
take
∼
the
“(
Y
G,
S
⊆
Node(
Y
G),
φ
:
Y
G
S
→
G)”
of
Definition
3.5
to
be
the
degeneration
structure
of
(a),
∼
(c)
a
PFC-admissible
isomorphism
Y
Π
2
→
Π
2
that
lifts
the
isomorphism
of
(b),
and
(d)
a
nodal
subgroup
Π
e
⊆
Y
Π
1
of
Y
Π
1
associated
to
a
[uniquely
determined
—
cf.
Remark
3.5.1]
node
e
of
Y
G
such
that
the
image
of
the
nodal
subgroup
Π
e
⊆
Y
Π
1
of
(d)
∼
via
the
isomorphism
Y
Π
1
→
Π
1
of
(b)
coincides
with
J
⊆
Π
1
.
We
shall
say
that
a
closed
subgroup
T
⊆
Π
2/1
of
Π
2/1
is
a
tripodal
subgroup
associated
to
J
if
T
coincides
—
relative
to
some
choice
of
data
(a),
(b),
(c),
(d)
as
above
[but
cf.
also
Re-
∼
mark
3.6.1!]
—
with
the
image,
via
the
lifting
Y
Π
2
→
Π
2
of
(c),
of
some
{1,
2}-tripod
in
Y
Π
2/1
⊆
Y
Π
2
[cf.
[CbTpII],
Def-
inition
3.3,
(i)]
arising
from
e
[cf.
[CbTpII],
Definition
3.7,
(i)],
and,
moreover,
the
centralizer
Z
Π
2
(T
)
maps
bijectively,
via
p
Π
2/1
:
Π
2
Π
1
,
onto
J
⊆
Π
1
[cf.
[CbTpII],
Lemma
3.11,
(iv),
(vii)].
(ii)
Let
T
⊆
Π
2/1
be
a
tripodal
subgroup
associated
to
J
[cf.
(i)].
Then
we
shall
refer
to
a
closed
subgroup
of
T
that
arises
from
a
nodal
(respectively,
cuspidal)
subgroup
contained
in
the
{1,
2}-tripod
in
Y
Π
2/1
⊆
Y
Π
2
of
(i)
as
a
lifting
cycle-subgroup
(respectively,
distinguished
cuspidal
subgroup)
of
T
[cf.
the
right-
hand
portion
of
Figure
1].
Remark
3.6.1.
Note
that,
in
the
situation
of
Definition
3.6,
(i),
it
fol-
lows
immediately
from
Lemma
3.1,
(ii)
[i.e.,
by
considering
the
gener-
ization
of
Y
G
with
respect
to
Node(
Y
G)
\
{e}
—
cf.
[CbTpI],
Definition
2.8],
together
with
the
computation
of
the
centralizer
given
in
[CbTpII],
Lemma
3.11,
(vii),
and
the
commensurable
terminality
of
J
⊆
Π
1
[cf.
[CmbGC],
Proposition
1.2,
(ii)],
that
the
Π
2/1
-conjugacy
class
of
a
tripodal
subgroup
T
is
completely
determined
by
the
cycle-subgroup
J
⊆
Π
1
.
Remark
3.6.2.
(i)
Suppose
that
we
are
in
the
situation
of
Definition
3.5,
(i).
Re-
call
the
module
Λ
G
,
i.e.,
the
cyclotome
associated
to
G,
defined
in
[CbTpI],
Definition
3.8,
(i).
Thus,
as
an
abstract
module,
COMBINATORIAL
ANABELIAN
TOPICS
IV
103
Σ
of
Z.
Recall,
fur-
Λ
G
is
isomorphic
to
the
pro-Σ
completion
Z
thermore,
from
[CbTpI],
Corollary
3.9,
(v),
(vi),
that
one
may
construct
a
natural,
functorial
{±}-orbit
of
isomorphisms
∼
Π
e
−→
Λ
Y
G
∼
—
where
Π
e
⊆
Y
Π
1
→
Π
Y
G
[cf.
[CbTpII],
Definition
3.1,
(ii)]
denotes
a
nodal
subgroup
associated
to
e.
Thus,
by
apply-
∼
ing
the
natural,
functorial
[outer]
isomorphisms
Λ
Y
G
→
Λ
Y
G
S
∼
[cf.
[CbTpI],
Corollary
3.9,
(i)]
and
Φ
−1
:
Π
Y
G
→
Π
Y
G
S
[cf.
Y
G
S
[CbTpI],
Definition
2.10],
together
with
the
[outer]
isomor-
∼
∼
phisms
Λ
Y
G
S
→
Λ
G
and
Π
Y
G
S
→
Π
G
induced
by
φ,
we
obtain
a
natural
{±}-orbit
of
isomorphisms
∼
J
−→
Λ
G
associated
to
the
cycle-subgroup
J
⊆
Π
1
.
Note
that
this
{±}-
orbit
of
isomorphisms
is
functorial
with
respect
to
automor-
phisms
α
of
Π
1
such
that
α(J)
=
J,
and,
moreover,
the
outer
automorphism
of
Π
G
obtained
by
forming
the
conjugate
of
α
∼
by
the
natural
outer
isomorphism
Π
1
→
Π
G
is
graphic
[cf.
the
equivalence
(i)
⇔
(iii)
of
Theorem
3.3].
In
this
context,
it
is
natural
to
refer
to
either
of
the
two
isomorphisms
in
this
{±}-orbit
as
an
orientation
on
the
cycle-subgroup
J.
(ii)
Now
suppose
that
we
are
in
the
situation
of
Definition
3.6,
(i),
(ii).
Then
let
us
observe
that
the
natural
outer
surjec-
∼
tion
Y
Π
2/1
Y
Π
{2}
→
Y
Π
1
determined
by
Y
p
Π
{1,2}/{2}
induces
a
natural
isomorphism
∼
Λ
Y
G
2∈{1,2},e
2
−→
Λ
Y
G
[cf.
[CbTpI],
Corollary
3.9,
(ii)],
where
we
write
e
2
∈
Y
2
(k)
for
a
k-valued
point
of
Y
2
that
lies,
relative
to
Y
p
log
2/1
,
over
the
k-valued
point
of
Y
determined
by
the
node
e.
Write
v
for
the
vertex
of
Y
G
2∈{1,2},e
2
that
gives
rise
to
the
tripodal
subgroup
T
⊆
Π
2/1
.
Thus,
we
have
a
natural
isomorphism
∼
Λ
v
−→
Λ
Y
G
2∈{1,2},e
2
[cf.
[CbTpI],
Corollary
3.9,
(ii)].
Now
suppose
that
e
∗
is
a
node
of
Y
G
2∈{1,2},e
2
that
abuts
to
v
and,
moreover,
gives
rise
to
a
lifting
cycle-subgroup
J
∗
⊆
T
of
the
tripodal
subgroup
T
.
Thus,
one
verifies
immediately
that
the
natural
outer
sur-
∼
jection
Π
2/1
Π
{2}
→
Π
1
determined
by
p
Π
{1,2}/{2}
induces
a
∼
natural
isomorphism
J
∗
→
J
[cf.
[CbTpII],
Lemma
3.6,
(iv)].
Let
Π
e
∗
⊆
Π
Y
G
2∈{1,2},e
2
be
a
nodal
subgroup
associated
to
e
∗
.
Then
the
[unique!]
branch
of
e
∗
that
abuts
to
v
determines
a
104
YUICHIRO
HOSHI
AND
SHINICHI
MOCHIZUKI
natural
isomorphism
∼
Π
e
∗
−→
Λ
v
[cf.
[CbTpI],
Corollary
3.9,
(v)].
Thus,
by
composing
the
iso-
morphisms
of
the
last
three
displays
with
the
isomorphism
∼
∼
Λ
Y
G
→
Λ
Y
G
S
→
Λ
G
discussed
in
(i)
and
the
inverse
of
the
∼
tautological
isomorphism
Π
e
∗
→
J
∗
,
we
obtain
a
natural
iso-
morphism
∼
J
∗
−→
Λ
G
associated
to
the
lifting
cycle-subgroup
J
∗
⊆
T
.
Note
that
this
natural
isomorphism
is
functorial
with
respect
to
FC-
admissible
automorphisms
α
2
of
Π
2
such
that
α
2
(J
∗
)
=
J
∗
,
α
2
(T
)
=
T
,
and,
moreover,
the
outer
automorphism
of
Π
G
ob-
tained
by
forming
the
conjugate,
by
the
natural
outer
isomor-
∼
phism
Π
1
→
Π
G
,
of
the
outer
automorphism
of
Π
1
determined
by
α
2
is
graphic
[cf.
the
equivalence
(i)
⇔
(iii)
of
Theorem
3.3;
[CbTpII],
Lemma
3.11,
(vii)].
Finally,
one
verifies
immediately
from
the
construction
of
the
isomorphisms
of
[CbTpI],
Corol-
∼
lary
3.9,
(v),
that
if
one
composes
this
isomorphism
J
∗
→
Λ
G
∼
with
the
inverse
of
the
natural
isomorphism
J
∗
→
J
discussed
∼
above,
then
the
resulting
isomorphism
J
→
Λ
G
is
an
orienta-
tion
on
the
cycle-subgroup
J,
in
the
sense
of
the
discussion
of
(i),
and,
moreover,
that,
if
we
define
an
orientation
on
the
tripodal
subgroup
T
to
be
a
choice
of
a
T
-conjugacy
class
of
lifting
cycle-subgroups
of
T
,
then
the
resulting
assignment
orientations
on
T
−→
orientations
on
J
is
a
bijection
[between
sets
of
cardinality
2].
Lemma
3.7
(Induced
outomorphisms
of
tripods).
In
the
situa-
tion
of
Lemma
3.1,
suppose
that
X
log
=
Y
log
.
Write
c
∈
Cusp(G
2∈{1,2},e
2
)
for
the
cusp
arising
from
the
diagonal
divisor
in
X
×
k
X.
Let
Π
c
⊆
Π
G
2∈{1,2},e
2
be
a
cuspidal
subgroup
of
Π
G
2∈{1,2},e
2
associated
to
c.
Write
def
α
v
=
T
Π
v
(α
2
)
∈
Out(Π
v
)
[cf.
Lemma
3.1,
(ii);
[CbTpII],
Theorem
3.16,
(i)]
for
the
result
of
applying
the
tripod
homomorphism
T
Π
v
to
α
2
.
[Thus,
it
follows
immediately
from
Lemma
3.1,
(ii),
that
α
v
∈
Out
C
(Π
v
).]
Suppose,
moreover,
that
the
following
condition
is
satisfied:
∼
(c)
The
cuspidal
subgroup
Π
c
⊆
Π
G
2∈{1,2},e
2
←
Π
2/1
is
contained
in
Π
v
.
Then
the
following
hold:
COMBINATORIAL
ANABELIAN
TOPICS
IV
105
(i)
Since
Π
v
may
be
regarded
as
the
“Π
1
”
that
occurs
in
the
case
where
we
take
“X
log
”
to
be
the
smooth
log
curve
associated
to
P
1
k
\
{0,
1,
∞}
[cf.
[CbTpII],
Remark
3.3.1],
there
exists
a
uniquely
determined
outomorphism
ι
∈
Out(Π
v
)
of
Π
v
that
arises
from
an
automorphism
of
P
1
k
\{0,
1,
∞}
over
k
and
induces
a
nontrivial
automorphism
of
the
set
N
(v).
Write
def
def
|α
v
|
=
α
v
∈
Out(Π
v
)
(respectively,
|α
v
|
=
ι◦α
v
∈
Out(Π
v
))
if
α
v
∈
Out
C
(Π
v
)
cusp
(respectively,
∈
Out
C
(Π
v
)
cusp
)
[cf.
[CbTpII],
Definition
3.4,
(i)].
Then
it
holds
that
|α
v
|
∈
Out
C
(Π
v
)
cusp
.
(ii)
Let
Π
tpd
⊆
Π
3
be
a
central
{1,
2,
3}-tripod
of
Π
3
[cf.
[CbTpII],
Definitions
3.3,
(i);
3.7,
(ii)].
Then
every
geometric
[cf.
∼
[CbTpII],
Definition
3.4,
(ii)]
outer
isomorphism
Π
tpd
→
Π
v
∼
satisfies
the
following
condition:
Let
β
∈
Out(Π
1
)
→
Out(Π
G
)
∼
be
an
outomorphism
of
Π
1
→
Π
G
that
is
group-theoretically
nodal
and
3-cuspidalizable,
i.e.,
β
∈
Out(Π
1
)
arises
from
a(n)
[uniquely
determined
—
cf.
[NodNon],
Theorem
B]
FC-
admissible
outomorphism
β
3
∈
Out
FC
(Π
3
).
Then
the
image
T
Π
tpd
(β
3
)
∈
Out(Π
tpd
)
[cf.
[CbTpII],
Definition
3.19]
coincides
—
relative
to
the
∼
outer
isomorphism
Π
tpd
→
Π
v
under
consideration
—
with
|β
v
|
∈
Out(Π
v
)
def
[cf.
(i)],
where
we
write
β
v
=
T
Π
v
(β
3
)
∈
Out(Π
v
).
In
particu-
lar,
it
holds
that
|β
v
|
∈
Out
C
(Π
v
)
Δ+
[cf.
[CbTpII],
Definition
3.4,
(i)].
Proof.
Assertion
(i)
follows
immediately
from
the
various
definitions
involved.
Next,
we
verify
assertion
(ii).
Let
us
first
observe
that
the
inclusion
|β
v
|
∈
Out
C
(Π
v
)
Δ
follows
immediately
from
the
coincidence
of
T
Π
tpd
(β
3
)
with
|β
v
|,
relative
to
some
specific
geometric
outer
iso-
∼
morphism
Π
tpd
→
Π
v
,
together
with
the
second
displayed
equality
of
[CbTpII],
Theorem
3.16,
(v).
The
inclusion
|β
v
|
∈
Out
C
(Π
v
)
Δ+
then
follows
from
[CbTpII],
Lemma
3.5;
[CbTpII],
Theorem
3.17,
(i)
[ap-
plied
in
the
case
where
we
take
the
“(Π
2
,
T,
T
)”
of
loc.
cit.
to
be
(Π
3/1
,
T,
Π
tpd
)].
Moreover,
it
follows
immediately
from
the
various
def-
initions
involved
that
the
inclusion
|β
v
|
∈
Out
C
(Π
v
)
Δ
allows
one
to
conclude
that
the
coincidence
of
T
Π
tpd
(β
3
)
with
|β
v
|,
relative
to
some
∼
specific
geometric
outer
isomorphism
Π
tpd
→
Π
v
,
implies
the
coinci-
dence
of
T
Π
tpd
(β
3
)
with
|β
v
|,
relative
to
an
arbitrary
geometric
outer
∼
isomorphism
Π
tpd
→
Π
v
.
Thus,
to
complete
the
verification
of
assertion
(ii),
it
suffices
to
verify
the
coincidence
of
T
Π
tpd
(β
3
)
with
|β
v
|,
relative
106
YUICHIRO
HOSHI
AND
SHINICHI
MOCHIZUKI
∼
to
the
specific
geometric
outer
isomorphism
Π
tpd
→
Π
v
whose
exis-
tence
is
guaranteed
by
[CbTpII],
Theorem
3.18,
(ii).
In
the
following
∼
discussion,
we
fix
this
specific
geometric
outer
isomorphism
Π
tpd
→
Π
v
.
Next,
let
us
observe
that
if
β
v
=
|β
v
|,
i.e.,
β
v
∈
Out
C
(Π
v
)
cusp
,
then
it
follows
immediately
from
[CbTpII],
Theorems
3.16,
(v);
3.18,
(ii),
that
T
Π
tpd
(β
3
)
∈
Out(Π
tpd
)
coincides
with
|β
v
|
∈
Out(Π
v
).
Thus,
to
complete
the
verification
of
assertion
(ii),
we
may
assume
without
loss
of
generality
that
β
v
=
|β
v
|,
i.e.,
that
β
v
∈
Out
C
(Π
v
)
cusp
.
Then
let
us
observe
that
collections
of
data
consisting
of
smooth
log
curves
that
[by
gluing
at
prescribed
cusps]
give
rise
to
a
stable
log
curve
whose
associated
semi-graph
of
anabelioids
[of
pro-Σ
PSC-type]
is
isomorphic
to
G
may
be
parametrized
by
a
smooth,
connected
moduli
stack.
Thus,
one
verifies
easily
that,
by
considering
the
étale
fundamental
groupoid
of
this
moduli
stack,
together
with
a
suitable
scheme-theoretic
auto-
morphism
of
order
2
of
a
collection
of
data
parametrized
by
this
mod-
uli
stack,
one
obtains
a
3-cuspidalizable
automorphism
ξ
∈
Aut(G)
(→
Out(Π
G
))
of
G
such
that
ξ
v
[i.e.,
the
“α
v
”
that
occurs
in
the
case
where
we
take
“α”
to
be
ξ]
coincides
with
ι.
Thus,
by
applying
the
portion
of
assertion
(ii)
that
has
already
been
verified
to
ξ
◦
β,
we
con-
clude
that,
to
complete
the
verification
of
assertion
(ii),
it
suffices
to
verify
that
T
Π
tpd
(ξ
3
)
=
1.
On
the
other
hand,
this
follows
immediately
from
the
fact
that
ξ
was
assumed
to
arise
from
a
scheme-theoretic
au-
tomorphism
[cf.
also
[CbTpII],
Theorem
3.16,
(v)].
This
completes
the
proof
of
assertion
(ii)
and
hence
of
Lemma
3.7.
Definition
3.8.
Let
J
⊆
Π
1
be
a
2-cuspidalizable
cycle-subgroup
[cf.
Definition
3.5,
(i),
(ii)];
let
us
fix
associated
data
as
in
Definition
3.6,
(i),
(a),
(b),
(c),
(d).
Relative
to
this
data,
suppose
that
T
⊆
Π
2/1
is
a
tripodal
subgroup
associated
to
J
⊆
Π
1
[cf.
Definition
3.6,
(i)],
and
that
I
⊆
T
is
a
distinguished
cuspidal
subgroup
of
T
[cf.
Definition
3.6,
(ii)].
Note
that
this
data,
together
with
the
log
scheme
structure
of
Y
log
,
allows
one
to
speak
of
geometric
[cf.
[CbTpII],
Definition
3.4,
(ii)]
out-
omorphisms
of
T
.
Then
one
verifies
easily
that
there
exists
a
uniquely
determined
nontrivial
geometric
outomorphism
of
T
that
preserves
the
T
-conjugacy
class
of
I.
Thus,
since
I
is
commensurably
terminal
in
T
[cf.
[CmbGC],
Proposition
1.2,
(ii)],
there
exists
a
uniquely
determined
I-conjugacy
class
of
automorphisms
of
T
that
lifts
this
outomorphism
and
preserves
I
⊆
T
.
We
shall
refer
to
this
I-conjugacy
class
of
auto-
morphisms
of
T
as
the
cycle
symmetry
associated
to
I.
Before
proceeding,
we
pause
to
observe
the
following
interesting
“al-
ternative
formulation”
of
the
essential
content
of
Lemma
3.7,
(ii).
COMBINATORIAL
ANABELIAN
TOPICS
IV
107
Lemma
3.9
(Geometricity
of
conjugates
of
geometric
outer
isomorphisms).
Suppose
that
we
are
in
the
situation
of
[CbTpII],
Theorem
3.18,
(ii),
i.e.,
n
≥
3,
and
T
(respectively,
T
)
is
an
E-
(respectively,
E
-)
tripod
of
Π
n
for
some
subset
E
⊆
{1,
.
.
.
,
n}
(re-
∼
spectively,
E
⊆
{1,
.
.
.
,
n}).
Let
φ
:
T
→
T
be
a
geometric
[cf.
[CbTpII],
Definition
3.4,
(ii)]
outer
isomorphism.
Then,
for
every
α
∈
Out
FC
(Π
n
)[T,
T
:
{|C|}],
the
composite
of
outer
isomorphisms
T
T
(α)
∼
φ
∼
T
−→
T
−→
T
T
T
(α)
−1
∼
−→
T
[cf.
[CbTpII],
Theorem
3.16,
(i)]
is
equal
to
φ.
Proof.
Let
us
first
observe
that
the
validity
of
Lemma
3.9
for
some
spe-
cific
geometric
outer
isomorphism
“φ”
follows
formally
from
the
com-
mutative
diagram
of
[CbTpII],
Theorem
3.18,
(ii).
Thus,
the
validity
of
Lemma
3.9
for
an
arbitrary
geometric
outer
isomorphism
“φ”
follows
immediately
from
the
equality
of
the
first
display
of
[CbTpII],
Theorem
3.18,
(i),
i.e.,
the
fact
that
T
T
(α)
commutes
with
arbitrary
geometric
outomorphisms
of
T
.
This
completes
the
proof
of
Lemma
3.9.
Remark
3.9.1.
One
verifies
immediately
that
a
similar
argument
to
the
argument
applied
in
the
proof
of
Lemma
3.9
yields
evident
ana-
logues
of
Lemma
3.9
in
the
respective
situations
of
[CbTpII],
Theorem
3.17,
(i),
(ii).
Theorem
3.10
(Canonical
liftings
of
cycles).
In
the
notation
of
the
discussion
at
the
beginning
of
the
present
§3,
let
I
⊆
Π
2/1
⊆
Π
2
be
a
cuspidal
inertia
group
associated
to
the
diagonal
cusp
of
a
fiber
of
p
log
2/1
;
Π
tpd
⊆
Π
3
a
3-central
{1,
2,
3}-tripod
of
Π
3
[cf.
[CbTpII],
Definition
3.7,
(ii)];
I
tpd
⊆
Π
tpd
a
cuspidal
subgroup
of
Π
tpd
that
does
∗
∗∗
not
arise
from
a
cusp
of
a
fiber
of
p
log
3/2
;
J
tpd
,
J
tpd
⊆
Π
tpd
cuspidal
∗
∗∗
subgroups
of
Π
tpd
such
that
I
tpd
,
J
tpd
,
and
J
tpd
determine
three
dis-
tinct
Π
tpd
-conjugacy
classes
of
closed
subgroups
of
Π
tpd
.
[Note
that
one
verifies
immediately
from
the
various
definitions
involved
that
such
∗
∗∗
cuspidal
subgroups
I
tpd
,
J
tpd
,
and
J
tpd
always
exist.]
For
positive
inte-
FC
gers
n
≥
2,
m
≤
n
and
α
∈
Aut
(Π
n
)
[cf.
[CmbCsp],
Definition
1.1,
(ii)],
write
α
m
∈
Aut
FC
(Π
m
)
for
the
automorphism
of
Π
m
determined
by
α;
Aut
FC
(Π
n
,
I)
⊆
Aut
FC
(Π
n
)
for
the
subgroup
consisting
of
β
∈
Aut
FC
(Π
n
)
such
that
β
2
(I)
=
I;
Aut
FC
(Π
n
)
G
⊆
Aut
FC
(Π
n
)
108
YUICHIRO
HOSHI
AND
SHINICHI
MOCHIZUKI
for
the
subgroup
consisting
of
β
∈
Aut
FC
(Π
n
)
such
that
the
image
of
β
via
the
composite
Aut
FC
(Π
n
)
Out
FC
(Π
n
)
→
Out
FC
(Π
1
)
→
Out(Π
G
)
—
where
the
second
arrow
is
the
natural
injection
of
[NodNon],
Theo-
rem
B,
and
the
third
arrow
is
the
homomorphism
induced
by
the
natural
∼
outer
isomorphism
Π
1
→
Π
G
—
is
graphic
[cf.
[CmbGC],
Definition
1.4,
(i)];
def
Aut
FC
(Π
n
,
I)
G
=
Aut
FC
(Π
n
,
I)
∩
Aut
FC
(Π
n
)
G
;
Cycle
n
(Π
1
)
for
the
set
of
n-cuspidalizable
cycle-subgroups
of
Π
1
[cf.
Defini-
tion
3.5,
(i),
(ii)];
Tpd
I
(Π
2/1
)
for
the
set
of
closed
subgroups
T
⊆
Π
2/1
such
that
T
is
a
tripodal
sub-
group
associated
to
some
2-cuspidalizable
cycle-subgroup
of
Π
1
[cf.
Definition
3.6,
(i)],
and,
moreover,
I
is
a
distinguished
cuspidal
subgroup
[cf.
Definition
3.6,
(ii)]
of
T
.
Then
the
following
hold:
(i)
Let
n
≥
2
be
a
positive
integer,
α
∈
Aut
FC
(Π
n
,
I)
G
,
J
∈
Cycle
n
(Π
1
),
and
T
∈
Tpd
I
(Π
2/1
).
Then
it
holds
that
α
1
(J)
∈
Cycle
n
(Π
1
),
α
2
(T
)
∈
Tpd
I
(Π
2/1
).
Thus,
Aut
FC
(Π
n
,
I)
G
acts
naturally
on
Cycle
n
(Π
1
),
Tpd
I
(Π
2/1
).
(ii)
Let
n
≥
2
be
a
positive
integer.
Then
there
exists
a
unique
Aut
FC
(Π
n
,
I)
G
-equivariant
[cf.
(i)]
map
C
I
:
Cycle
n
(Π
1
)
−→
Tpd
I
(Π
2/1
)
such
that,
for
every
J
∈
Cycle
n
(Π
1
),
C
I
(J)
is
a
tripodal
sub-
group
associated
to
J.
Moreover,
for
every
α
∈
Aut
FC
(Π
n
,
I)
G
∼
and
J
∈
Cycle
n
(Π
1
),
the
isomorphism
C
I
(J)
→
C
I
(α
1
(J))
induced
by
α
2
maps
every
lifting
cycle-subgroup
[cf.
Def-
inition
3.6,
(ii)]
of
C
I
(J)
bijectively
onto
a
lifting
cycle-
subgroup
of
C
I
(α
1
(J)).
(iii)
Let
n
≥
3
be
a
positive
integer.
Then
there
exists
an
assign-
ment
Cycle
n
(Π
1
)
J
→
syn
I,J
—
where
syn
I,J
denotes
an
I-conjugacy
class
of
isomorphisms
∼
Π
tpd
→
C
I
(J)
—
such
that
(a)
syn
I,J
maps
I
tpd
bijectively
onto
I,
∗
∗∗
(b)
syn
I,J
maps
the
subgroups
J
tpd
,
J
tpd
bijectively
onto
lift-
ing
cycle-subgroups
of
C
I
(J),
and
COMBINATORIAL
ANABELIAN
TOPICS
IV
109
(c)
for
α
∈
Aut
FC
(Π
n
,
I)
G
,
the
diagram
[of
I
tpd
-,
I-conjugacy
classes
of
isomorphisms]
Π
tpd
−−−→
⏐
⏐
syn
I,J
Π
tpd
⏐
⏐
syn
I,α
(J)
1
C
I
(J)
−−−→
C
I
(α
1
(J))
—
where
the
upper
horizontal
arrow
is
the
[uniquely
de-
termined
—
cf.
the
commensurable
terminality
of
I
tpd
in
Π
tpd
discussed
in
[CmbGC],
Proposition
1.2,
(ii)]
I
tpd
-
conjugacy
class
of
automorphisms
of
Π
tpd
that
lifts
T
Π
tpd
(α)
[cf.
[CbTpII],
Definition
3.19]
and
preserves
I
tpd
;
the
lower
horizontal
arrow
is
the
I-conjugacy
class
of
isomorphisms
induced
by
α
2
[cf.
(ii)]
—
commutes
up
to
possible
com-
position
with
the
cycle
symmetry
of
C
I
(α
1
(J))
associ-
ated
to
I
[cf.
Definition
3.8].
Finally,
the
assignment
J
→
syn
I,J
is
uniquely
determined,
up
to
possible
composition
with
cy-
cle
symmetries,
by
these
conditions
(a),
(b),
and
(c).
(iv)
Let
n
≥
3
be
a
positive
integer,
α
∈
Aut
FC
(Π
n
,
I)
G
,
and
J
∈
Cycle
n
(Π
1
).
Suppose
that
one
of
the
following
conditions
is
satisfied:
(a)
The
FC-admissible
outomorphism
of
Π
3
determined
by
α
3
is
∈
Out
FC
(Π
3
)
geo
[cf.
[CbTpII],
Definition
3.19].
(b)
Cusp(G)
=
∅.
(c)
n
≥
4.
Then
there
exists
an
automorphism
β
∈
Aut
FC
(Π
n
,
I)
G
such
that
the
FC-admissible
outomorphism
of
Π
3
determined
by
β
3
is
contained
in
Out
FC
(Π
3
)
geo
,
and,
moreover,
α
1
(J)
=
β
1
(J).
Finally,
the
diagram
[of
I
tpd
-,
I-conjugacy
classes
of
isomor-
phisms]
Π
tpd
⏐
⏐
syn
I,J
Π
tpd
⏐
⏐
syn
I,α
(J)
=syn
I,β
(J)
1
1
C
I
(J)
−−−→
C
I
(α
1
(J))
=
C
I
(β
1
(J))
—
where
the
lower
horizontal
arrow
is
the
isomorphism
induced
by
β
2
[cf.
(ii)]
—
commutes
up
to
possible
composition
with
the
cycle
symmetry
of
C
I
(α
1
(J))
=
C
I
(β
1
(J))
associated
to
I.
Proof.
Assertion
(i)
follows
immediately
from
the
various
definitions
involved.
Next,
we
verify
assertion
(ii).
The
initial
portion
of
assertion
(ii)
follows
immediately
from
the
discussion
of
Remark
3.6.1,
together
110
YUICHIRO
HOSHI
AND
SHINICHI
MOCHIZUKI
with
the
fact
that
T
is
uniquely
determined
among
its
Π
2/1
-conjugates
by
the
condition
I
⊆
T
[cf.
[CmbGC],
Proposition
1.5,
(i)].
The
final
portion
of
assertion
(ii)
follows
immediately
from
Lemma
3.1,
(ii)
[i.e.,
by
considering
a
suitable
generization
operation,
as
in
the
discussion
of
Remark
3.6.1].
This
completes
the
proof
of
assertion
(ii).
Next,
we
verify
assertion
(iii).
Let
us
fix
data
∼
∼
(
Y
G,
S
⊆
Node(
Y
G),
φ
:
Y
G
S
→
G);
Y
Π
1
→
Π
1
;
Y
∼
Π
2
→
Π
2
;
Π
e
⊆
Y
Π
1
for
J
∈
Cycle
n
(Π
1
)
as
in
Definition
3.6,
(i),
(a),
(b),
(c),
(d),
and
let
Y
T
⊆
Y
Π
2/1
be
a
{1,
2}-tripod
as
in
the
discussion
of
Definition
3.6,
(i).
Let
Y
Π
tpd
⊆
Y
Π
3
be
a
3-central
tripod
of
Y
Π
3
.
Here,
we
note
that
since
J
∈
Cycle
n
(Π
1
),
and
n
≥
3,
it
follows
that
the
above
isomorphism
∼
∼
Y
Π
2
→
Π
2
lifts
to
a
PFC-admissible
isomorphism
Y
Π
3
→
Π
3
that
maps
Y
Π
tpd
to
a
Π
3
-conjugate
of
Π
tpd
[cf.
[NodNon],
Theorem
B;
[CbTpII],
Theorem
3.16,
(v);
[CbTpII],
Remark
4.14.1].
Now
one
verifies
immediately
that,
to
verify
the
existence
portion
of
assertion
(iii),
by
applying
a
suitable
generization
operation
as
in
the
discussion
of
Remark
3.6.1,
we
may
assume
without
loss
of
gen-
erality
that
Node(
Y
G)
=
1
[an
assumption
that
will
be
invoked
when
we
apply
Lemma
3.7
in
the
argument
to
follow].
Then,
by
considering
the
geometric
[hence,
in
particular,
C-admissible]
outer
isomorphism
of
[CbTpII],
Theorem
3.18,
(ii),
in
the
case
where
we
take
the
“(T,
T
)”
of
[CbTpII],
Theorem
3.18,
(ii),
to
be
(
Y
Π
tpd
,
Y
T
),
we
obtain
an
outer
∼
isomorphism
Π
tpd
→
C
I
(J).
Moreover,
by
considering
the
composite
of
this
outer
isomorphism
with
a
suitable
geometric
outomorphism
of
Π
tpd
,
we
may
assume
without
loss
of
generality
that
this
outer
iso-
∼
morphism
Π
tpd
→
C
I
(J)
maps
the
Π
tpd
-conjugacy
class
of
I
tpd
to
the
C
I
(J)-conjugacy
class
of
I.
Thus,
since
I
is
commensurably
terminal
in
C
I
(J)
[cf.
[CmbGC],
Proposition
1.2,
(ii)],
we
obtain
a
uniquely
de-
∼
termined
I-conjugacy
class
of
isomorphisms
syn
I,J
:
Π
tpd
→
C
I
(J)
that
lifts
the
outer
isomorphism
just
discussed
and
satisfies
condition
(a).
On
the
other
hand,
one
verifies
immediately
from
the
various
definitions
involved
that
syn
I,J
also
satisfies
condition
(b).
Next,
we
verify
that
syn
I,J
satisfies
condition
(c).
To
this
end,
let
us
observe
that
it
follows
immediately
from
the
various
definitions
involved
[cf.
also
our
assumption
that
Node(
Y
G)
=
1],
that
α
1
(J)
admits
data
as
in
Definition
3.6,
(i),
(a),
(b),
(c),
(d),
such
that
•
the
portion
of
this
data
that
corresponds
to
the
data
of
Defi-
nition
3.6,
(i),
(a),
(d),
is
of
the
form
∼
(
Y
G,
S
⊆
Node(
Y
G),
ψ
:
Y
G
S
→
G);
Π
e
⊆
Y
Π
1
∼
for
some
isomorphism
ψ
:
Y
G
S
→
G,
and,
moreover,
COMBINATORIAL
ANABELIAN
TOPICS
IV
111
•
the
composite
Y
∼
α
2
∼
∼
Π
2
−→
Π
2
−→
Π
2
←−
Y
Π
2
—
where
the
first
(respectively,
third)
arrow
is
the
isomor-
phism
arising
from
the
data
[cf.
Definition
3.6,
(i),
(c)]
for
J
(respectively,
α
1
(J))
∈
Cycle
n
(Π
1
)
under
consideration
—
is
the
identity
automorphism.
Thus,
to
verify
the
assertion
that
syn
I,J
satisfies
condition
(c),
it
suffices
∼
to
verify
that
the
I-conjugacy
class
of
isomorphisms
“syn
I,J
:
Π
tpd
→
C
I
(J)”
constructed
above
from
a
fixed
choice
of
data
as
in
Defini-
tion
3.6,
(i),
(a),
(b),
(c),
(d)
does
not
depend
on
this
choice
of
data.
On
the
other
hand,
this
follows
immediately
from
Lemma
3.7,
(ii)
[cf.
our
assumption
that
Node(
Y
G)
=
1].
Finally,
we
consider
the
final
portion
of
assertion
(iii)
concerning
uniqueness.
To
this
end,
we
observe
that,
by
considering
the
case
where
Y
G,
as
well
as
each
of
the
branches
of
the
underlying
semi-graph
of
Y
G,
is
defined
over
a
number
field
F
,
it
follows
immediately,
by
considering
automorphisms
α
∈
Aut
FC
(Π
n
,
I)
G
that
arise
from
scheme
theory,
that
given
any
element
γ
∈
Out(Π
tpd
)
that
arises
from
an
element
of
the
absolute
Galois
group
of
F
,
there
exists
an
α
∈
Aut
FC
(Π
n
,
I)
G
such
that
α(J)
=
J
and
T
Π
tpd
(α)
=
γ.
Thus,
the
uniqueness
under
consideration
follows
immediately
from
the
geometricity
of
elements
of
Out(Π
tpd
)
that
commute
with
the
image
of
the
absolute
Galois
group
of
F
,
i.e.,
in
other
words,
from
the
Grothendieck
Conjecture
for
tripods
over
number
fields
[cf.
[Tama1],
Theorem
0.3;
[LocAn],
Theorem
A].
This
completes
the
proof
of
assertion
(iii).
Finally,
we
verify
assertion
(iv).
If
condition
(a)
is
satisfied,
then,
by
taking
the
“β”
of
assertion
(iv)
to
be
α,
we
conclude
that
assertion
(iv)
follows
immediately
from
assertion
(iii),
together
with
the
definition
of
Out
FC
(Π
n
)
geo
.
Next,
let
us
observe
that,
by
applying
assertion
(iv)
in
the
case
where
condition
(a)
is
satisfied,
we
conclude
that,
to
verify
assertion
(iv)
in
the
case
where
either
(b)
or
(c)
is
satisfied,
it
suffices
to
verify
that
the
following
assertion
holds:
Claim
3.10.A:
Write
Out(Π
1
⊇
J)
⊆
Out(Π
1
)
for
the
subgroup
of
Out(Π
1
)
consisting
of
outomor-
phisms
of
Π
1
that
preserve
the
Π
1
-conjugacy
class
of
J
and
def
Out
FC
(Π
n
)
G
=
Aut
FC
(Π
n
)
G
/Inn(Π
n
)
⊆
Out
FC
(Π
n
).
Then
every
element
of
the
image
of
the
natural
injec-
tion
Out
FC
(Π
n
)
G
→
Out
FC
(Π
1
)
112
YUICHIRO
HOSHI
AND
SHINICHI
MOCHIZUKI
[cf.
[NodNon],
Theorem
B]
may
be
written
as
a
prod-
uct
of
an
element
of
the
image
of
the
natural
injec-
tion
Out
FC
(Π
n
)
geo
→
Out
FC
(Π
1
)
and
an
element
of
def
Out(Π
1
⊇
J)
G
=
Out(Π
1
⊇
J)
∩
Out
FC
(Π
1
)
G
.
To
verify
Claim
3.10.A,
write
Out
FC
(Π
n
,
J)
G
⊆
Out
FC
(Π
n
)
G
for
the
subgroup
of
Out
FC
(Π
n
)
G
obtained
by
forming
the
inverse
image
of
the
closed
subgroup
Out(Π
1
⊇
J)
⊆
Out(Π
1
)
via
the
natural
injection
Out
FC
(Π
n
)
G
→
Out
FC
(Π
1
).
Then
one
verifies
immediately,
by
consid-
ering
the
exact
sequence
T
Πtpd
1
−→
Out
FC
(Π
n
)
geo
−→
Out
FC
(Π
n
)
−→
Out
C
(Π
tpd
)
Δ+
−→
1
[cf.
conditions
(b),
(c);
[CbTpII],
Definition
3.19;
[CbTpII],
Corollary
4.15],
that,
to
verify
Claim
3.10.A,
it
suffices
to
verify
that
the
following
assertion
holds:
Claim
3.10.B:
The
composite
T
Πtpd
Out
FC
(Π
n
,
J)
G
→
Out
FC
(Π
n
)
Out
C
(Π
tpd
)
Δ+
is
surjective.
∼
To
verify
Claim
3.10.B,
let
(
Y
G,
S
⊆
Node(
Y
G),
φ
:
Y
G
S
→
G)
be
an
n-cuspidalizable
degeneration
structure
on
G
with
respect
to
which
J
is
a
cycle-subgroup
such
that
Y
G
is
totally
degenerate
[cf.
[CbTpI],
Defini-
tion
2.3,
(iv)].
[One
verifies
immediately
that
such
a
degeneration
struc-
ture
always
exists.]
Now
let
us
identify
Out
FC
(Π
n
)
with
Out
FC
(
Y
Π
n
)
via
a(n)
[uniquely
determined,
up
to
permutation
of
the
n
factors
—
cf.
[NodNon],
Theorem
B]
PFC-admissible
[cf.
[CbTpI],
Definition
1.4,
∼
(iii)]
outer
isomorphism
Π
n
→
Y
Π
n
that
is
compatible
with
the
out-
omorphism
of
the
display
of
Definition
3.5,
(i)
[cf.
[CbTpII],
Propo-
sition
3.24,
(i)].
Then
it
follows
immediately
from
the
various
defini-
tions
involved
that
the
closed
subgroup
Out
FC
(
Y
Π
n
)
brch
⊆
Out
FC
(
Y
Π
n
)
[cf.
[CbTpII],
Definition
4.6,
(i)]
is
contained
in
the
closed
subgroup
Out
FC
(Π
n
,
J)
G
⊆
Out
FC
(Π
n
).
On
the
other
hand,
it
follows
immedi-
ately
from
the
proof
of
[CbTpII],
Corollary
4.15,
that
the
composite
T
Πtpd
Out
FC
(
Y
Π
n
)
brch
→
Out
FC
(
Y
Π
n
)
=
Out
FC
(Π
n
)
Out
C
(Π
tpd
)
Δ+
is
surjective.
This
completes
the
proof
of
Claim
3.10.B,
hence
also
of
assertion
(iv)
in
the
case
where
either
(b)
or
(c)
is
satisfied.
Remark
3.10.1.
(i)
The
content
of
Theorem
3.10,
(iv),
may
be
regarded,
i.e.,
by
considering
the
various
lifting
cycle-subgroups
involved,
as
a
formulation
of
the
construction
of
the
two
sections
discussed
in
[Bgg2],
Proposition
2.7
[which
plays
an
essential
role
in
the
COMBINATORIAL
ANABELIAN
TOPICS
IV
113
proof
of
[Bgg2],
Theorem
2.4],
in
terms
of
the
purely
combina-
torial
and
algebraic
techniques
developed
in
the
present
series
of
papers.
(ii)
In
this
context,
we
observe
in
passing
that
[one
verifies
imme-
diately
that]
for
arbitrary
nonnegative
integers
g,
r
such
that
•
3g
−
3
+
r
>
0,
and,
moreover,
•
if
g
=
0,
then
r
is
even,
there
exists
a
stable
log
curve
of
type
(g,
r)
which
admits
an
automorphism
that
is
linear
over
the
base
scheme
under
con-
sideration
and
fixes
a
node
of
the
stable
log
curve,
but
switches
the
branches
of
this
node.
Thus,
by
considering
the
resulting
automorphism
of
the
associated
semi-graph
of
anabelioids
of
pro-Σ
PSC-type,
one
concludes
that
the
diagrams
of
Theo-
rem
3.10,
(iii),
(iv),
fail
to
commute,
in
general,
if
one
does
not
allow
for
the
possibility
of
composition
with
a
cycle
symmetry.
This
situation
contrasts
with
the
situation
discussed
in
[Bgg2],
Proposition
2.7,
where
two
independent
sections
are
obtained,
by
considering
orientations
on
the
various
cycles
involved.
(iii)
The
orientation-theoretic
portion
of
[Bgg2],
Proposition
2.7,
referred
to
in
(ii)
above
may
be
interpreted,
from
the
point
of
view
of
the
present
paper,
as
a
lifting
“C
±
I
”
of
the
map
C
I
of
Theorem
3.10,
(ii),
as
follows.
In
the
notation
of
Theorem
3.10,
let
us
write
•
Cycle
n
(Π
1
)
±
for
the
set
of
pairs
consisting
of
a
cycle-
subgroup
J
∈
Cycle
n
(Π
1
)
and
an
orientation
on
J
[cf.
Remark
3.6.2,
(i)];
•
Tpd
I
(Π
2/1
)
±
for
the
set
of
pairs
consisting
of
a
tripodal
subgroup
T
∈
Tpd
I
(Π
2/1
)
and
an
orientation
on
T
[cf.
Remark
3.6.2,
(ii)].
Thus,
one
has
natural
surjections
Cycle
n
(Π
1
)
±
Cycle
n
(Π
1
),
Tpd
I
(Π
2/1
)
±
Tpd
I
(Π
2/1
),
which
may
be
regarded
as
torsors
over
the
group
{±1}.
Moreover,
one
verifies
immediately
from
the
functoriality
of
the
various
isomorphisms
that
appeared
in
the
constructions
of
Remark
3.6.2,
(i),
(ii),
that
the
action
[cf.
Theorem
3.10,
(i)]
of
Aut
FC
(Π
n
,
I)
G
on
the
sets
Cycle
n
(Π
1
),
Tpd
I
(Π
2/1
)
lifts
naturally
to
an
action
of
Aut
FC
(Π
n
,
I)
G
on
the
sets
Cycle
n
(Π
1
)
±
,
Tpd
I
(Π
2/1
)
±
.
Thus,
the
inverse
of
the
bijective
correspondence
of
the
final
display
of
Remark
3.6.2,
(ii),
determines
a
natural
Aut
FC
(Π
n
,
I)
G
-equivariant
lift-
ing
n
±
C
±
−→
Tpd
I
(Π
2/1
)
±
I
:
Cycle
(Π
1
)
of
the
map
C
I
of
Theorem
3.10,
(ii).
[Thus,
the
Aut
FC
(Π
n
,
I)
G
-
±
equivariance
of
C
±
I
implies,
in
particular,
that
C
I
does
not
fac-
n
tor
through
the
natural
surjection
Cycle
(Π
1
)
±
Cycle
n
(Π
1
).]
Moreover,
if
n
≥
3,
and
one
regards
the
Π
tpd
-conjugacy
class
114
YUICHIRO
HOSHI
AND
SHINICHI
MOCHIZUKI
∗
of
cuspidal
subgroups
of
Π
tpd
determined
by
J
tpd
as
being
“positive”,
then
it
follows
immediately
from
the
definition
of
Tpd
I
(Π
2/1
)
±
that
this
lifting
C
±
I
naturally
determines
an
as-
signment
Cycle
n
(Π
1
)
±
J
±
→
syn
±
I,J
±
—
where
J
±
→
J
∈
Cycle
n
(Π
1
),
and
syn
±
I,J
±
denotes
an
I-
∼
conjugacy
class
of
isomorphisms
Π
tpd
→
C
I
(J)
that
coincides,
up
to
possible
composition
with
a
cycle
symmetry,
with
the
I-
conjugacy
class
of
isomorphisms
syn
I,J
of
Theorem
3.10,
(iii)
—
such
that
if,
in
the
diagram
[of
I
tpd
-,
I-conjugacy
classes
of
isomorphisms]
in
the
display
of
Theorem
3.10,
(iii),
(c),
one
replaces
“syn”
by
“syn
±
”,
then
the
diagram
commutes,
i.e.,
even
if
one
does
not
allow
for
possible
composition
with
cycle
symmetries.
Definition
3.11.
Suppose
that
Σ
=
Primes,
and
that
k
=
C,
i.e.,
that
we
are
in
the
situation
of
Definition
2.22.
We
shall
apply
the
notational
conventions
established
in
Definition
2.22.
Moreover,
we
shall
use
similar
notation
def
def
def
def
=
π
1
(Y
E
),
Y
n
=
Y
{1,...,n}
,
Y
=
Y
1
,
Y
E
=
(Y
E
log
)
an
(C)|
s
,
Y
Π
disc
E
Y
def
disc
Y
an
Y
disc
Y
disc
Π
disc
=
Y
Π
disc
p
E/E
:
Y
E
→
Y
E
,
Y
p
Π
{1,...,n}
,
E/E
:
Π
E
Π
E
,
n
Y
def
disc
Y
Π
Y
disc
Π
disc
E/E
=
Ker(
p
E/E
)
⊆
Π
E
,
Y
an
def
Y
an
p
n/m
=
p
{1,...,n}/{1,...,m}
:
Y
n
−→
Y
m
,
Y
Π
disc
def
Y
Π
disc
p
n/m
=
p
{1,...,n}/{1,...,m}
:
Y
Π
disc
Y
Π
disc
n
m
,
Y
Y
disc
Y
disc
Y
disc
Π
(−)
,
Π
disc
n/m
=
Π
{1,...,n}/{1,...,m}
⊆
Π
n
,
def
Y
disc
G
disc
,
Y
G
i∈E,y
,
Π
Y
G
disc
,
Π
Y
G
i∈E,y
disc
for
objects
associated
to
the
stable
log
curve
Y
log
=
Y
1
log
to
the
notation
introduced
in
Definitions
2.22,
2.23.
Definition
3.12.
Let
J
be
a
semi-graph
of
temperoids
of
HSD-type
[cf.
Definition
2.3,
(iii)].
Then
we
shall
refer
to
a
triple
∼
(H,
S
⊆
Node(H),
φ
:
H
S
→
J
)
[cf.
Definition
2.9]
consisting
of
a
semi-graph
of
temperoids
of
HSD-
∼
type
H,
a
subset
S
⊆
Node(H),
and
an
isomorphism
φ
:
H
S
→
J
of
semi-graphs
of
temperoids
of
HSD-type
as
a
degeneration
structure
on
J
[cf.
[CbTpII],
Definition
3.23,
(i)].
COMBINATORIAL
ANABELIAN
TOPICS
IV
115
Definition
3.13.
In
the
situation
of
Definition
3.11:
disc
∼
(i)
Let
(
Y
G
disc
,
S
⊆
Node(
Y
G
disc
),
φ
:
Y
G
S
→
G
disc
)
be
a
degen-
eration
structure
on
G
disc
[cf.
Definition
3.12],
e
∈
S,
and
a
subgroup
of
Π
disc
J
⊆
Π
disc
1
1
.
Then
we
shall
say
that
J
⊆
disc
Π
1
is
a
cycle-subgroup
of
Π
disc
[with
respect
to
(
Y
G
disc
,
S
⊆
1
∼
disc
Node(
Y
G
disc
),
φ
:
Y
G
S
→
G
disc
),
associated
to
e
∈
S]
if
J
is
disc
contained
in
the
Π
1
-conjugacy
class
of
subgroups
of
Π
disc
ob-
1
tained
by
forming
the
image
of
a
nodal
subgroup
of
Π
Y
G
disc
associated
to
e
via
the
composite
of
outer
isomorphisms
Φ
−1
Y
disc
G
S
∼
∼
∼
disc
Π
Y
G
disc
−→
Π
Y
G
S
disc
−→
Π
G
disc
−→
Π
1
—
where
the
first
arrow
is
the
inverse
of
the
specialization
outer
isomorphism
Φ
Y
G
S
disc
[cf.
Proposition
2.10],
the
second
ar-
row
is
the
graphic
[cf.
Definition
2.7,
(ii)]
outer
isomorphism
∼
Π
Y
G
S
disc
→
Π
G
disc
induced
by
φ,
and
the
third
arrow
is
the
nat-
∼
of
[the
second
to
last
ural
outer
isomorphism
Π
G
disc
→
Π
disc
1
display
of]
Definition
2.23,
(i)
[cf.
the
left-hand
portion
of
Fig-
ure
1].
(ii)
Let
J
⊆
Π
1
disc
be
a
cycle-subgroup
of
Π
disc
[cf.
(i)].
Thus,
we
1
have
disc
∼
(a)
a
degeneration
structure
(
Y
G
disc
,
S
⊆
Node(
Y
G
disc
),
φ
:
Y
G
S
→
G
disc
)
on
G
disc
[cf.
Definition
3.12],
∼
(b)
an
isomorphism
Y
Π
disc
→
Π
disc
that
is
compatible
with
the
1
1
composite
of
the
display
of
(i)
[cf.
also
[the
second
to
last
display
of]
Definition
2.23,
(i)]
in
the
case
where
we
take
disc
∼
the
“(
Y
G
disc
,
S
⊆
Node(
Y
G
disc
),
φ
:
Y
G
S
→
G
disc
)”
of
(i)
to
be
the
degeneration
structure
of
(a),
∼
(c)
an
isomorphism
Y
Π
disc
→
Π
disc
that
lifts
[cf.
Corollary
2.20,
2
2
(v)]
the
isomorphism
of
(b)
and,
moreover,
determines
a
PFC-admissible
isomorphism
between
the
respective
profi-
nite
completions,
and
(d)
a
nodal
subgroup
Π
e
⊆
Y
Π
disc
of
Y
Π
disc
associated
to
a
1
1
[uniquely
determined
—
cf.
Corollary
2.18,
(iii)]
node
e
of
Y
disc
G
such
that
the
image
of
the
nodal
subgroup
Π
e
⊆
Y
Π
1
disc
of
∼
→
Π
disc
of
(b)
coincides
with
(d)
via
the
isomorphism
Y
Π
disc
1
1
disc
.
We
shall
say
that
a
subgroup
T
⊆
Π
disc
J
⊆
Π
disc
1
2/1
of
Π
2/1
is
a
tripodal
subgroup
associated
to
J
if
T
coincides
—
rela-
tive
to
some
choice
of
data
(a),
(b),
(c),
(d)
as
above
[but
cf.
also
Remark
3.6.1
and
Corollary
2.19,
(i)!]
—
with
the
image,
∼
of
(c),
of
some
{1,
2}-tripod
in
→
Π
disc
via
the
lifting
Y
Π
disc
2
2
Y
disc
Π
2/1
⊆
Y
Π
disc
[cf.
Definition
2.23,
(ii)]
arising
from
e
[cf.
Def-
2
inition
2.23,
(iii);
[CbTpII],
Definition
3.7,
(i)],
and,
moreover,
116
YUICHIRO
HOSHI
AND
SHINICHI
MOCHIZUKI
disc
disc
the
centralizer
Z
Π
disc
(T
)
maps
bijectively,
via
p
Π
2/1
:
Π
2
2
disc
disc
Π
1
,
onto
J
⊆
Π
1
[cf.
Corollary
2.17,
(i);
[CbTpII],
Lemma
3.11,
(iv),
(vii)].
(iii)
Let
J
⊆
Π
disc
be
a
cycle-subgroup
of
Π
disc
[cf.
(i)]
and
T
⊆
Π
disc
1
1
2/1
a
tripodal
subgroup
associated
to
J
[cf.
(ii)].
Then
we
shall
refer
to
a
subgroup
of
T
that
arises
from
a
nodal
(respectively,
cusp-
Y
disc
idal)
subgroup
contained
in
the
{1,
2}-tripod
in
Y
Π
disc
2/1
⊆
Π
2
of
(ii)
as
a
lifting
cycle-subgroup
(respectively,
distinguished
cuspidal
subgroup)
of
T
[cf.
the
right-hand
portion
of
Figure
1].
(iv)
Let
J
⊆
Π
disc
be
a
cycle-subgroup
[cf.
(i)];
T
⊆
Π
disc
1
2/1
a
tripodal
subgroup
associated
to
J
[cf.
(ii)];
I
⊆
T
a
distinguished
cuspi-
dal
subgroup
of
T
[cf.
(iii)].
Then
it
follows
immediately
from
the
various
definitions
involved,
together
with
Theorem
2.24,
(i),
that
there
exists
a
unique
outomorphism
ι
of
T
such
that
the
induced
outomorphism
of
the
profinite
completion
T
of
T
coincides
with
the
outomorphism
of
T
determined
by
the
cycle
symmetry
of
T
associated
to
the
profinite
completion
I
of
I
[cf.
Definition
3.8].
Moreover,
since
I
is
commensurably
terminal
in
T
[cf.
Corollary
2.18,
(v)],
it
follows
immediately
from
Corollary
2.17,
(ii),
that
there
exists
a
uniquely
deter-
mined
I-conjugacy
class
of
automorphisms
of
T
that
lifts
ι
and
preserves
I
⊆
T
.
We
shall
refer
to
this
I-conjugacy
class
of
automorphisms
of
T
as
the
cycle
symmetry
of
T
associated
to
I.
Theorem
3.14
(Discrete
version
of
canonical
liftings
of
cycles).
disc
In
the
notation
of
Definition
3.11,
let
I
⊆
Π
disc
be
a
cusp-
2/1
⊆
Π
2
idal
inertia
group
associated
to
the
diagonal
cusp
of
a
fiber
of
p
an
2/1
;
disc
disc
Π
tpd
⊆
Π
3
a
3-central
{1,
2,
3}-tripod
of
Π
3
[cf.
Definition
2.23,
(ii),
(iii)];
I
tpd
⊆
Π
tpd
a
cuspidal
subgroup
of
Π
tpd
that
does
not
arise
∗
∗∗
from
a
cusp
of
a
fiber
of
p
an
3/2
;
J
tpd
,
J
tpd
⊆
Π
tpd
cuspidal
subgroups
∗
∗∗
,
and
J
tpd
determine
three
distinct
Π
tpd
-
of
Π
tpd
such
that
I
tpd
,
J
tpd
conjugacy
classes
of
subgroups
of
Π
tpd
.
[Note
that
one
verifies
immedi-
ately
from
the
various
definitions
involved
that
such
cuspidal
subgroups
∗
∗∗
I
tpd
,
J
tpd
,
and
J
tpd
always
exist.]
For
α
∈
Aut
FC
(Π
disc
2
)
[cf.
the
nota-
tional
conventions
introduced
in
the
statement
of
Corollary
2.20],
write
α
1
∈
Aut
FC
(Π
disc
1
)
determined
by
α;
for
the
automorphism
of
Π
disc
1
FC
disc
Aut
FC
(Π
disc
2
,
I)
⊆
Aut
(Π
2
)
for
the
subgroup
consisting
of
β
∈
Aut
FC
(Π
disc
2
)
such
that
β(I)
=
I;
FC
G
disc
Aut
FC
(Π
disc
2
)
⊆
Aut
(Π
2
)
COMBINATORIAL
ANABELIAN
TOPICS
IV
117
for
the
subgroup
consisting
of
β
∈
Aut
FC
(Π
disc
2
)
such
that
the
image
of
∼
FC
FC
FC
disc
disc
β
via
the
composite
Aut
(Π
2
)
Out
(Π
disc
2
)
→
Out
(Π
1
)
→
Out(Π
G
disc
)
—
where
the
second
arrow
is
the
natural
bijection
of
Corol-
lary
2.20,
(v),
and
the
third
arrow
is
the
homomorphism
induced
by
∼
the
natural
outer
isomorphism
Π
disc
→
Π
G
disc
—
is
graphic
[cf.
Defi-
1
nition
2.7,
(ii)];
def
FC
G
disc
G
Aut
FC
(Π
disc
=
Aut
FC
(Π
disc
2
,
I)
2
,
I)
∩
Aut
(Π
2
)
;
Cycle(Π
disc
1
)
[cf.
Definition
3.13,
(i)];
for
the
set
of
cycle-subgroups
of
Π
disc
1
Tpd
I
(Π
disc
2/1
)
for
the
set
of
subgroups
T
⊆
Π
disc
2/1
such
that
T
is
a
tripodal
subgroup
[cf.
Definition
3.13,
(ii)],
associated
to
some
cycle-subgroup
of
Π
disc
1
and,
moreover,
I
is
a
distinguished
cuspidal
subgroup
[cf.
Defini-
tion
3.13,
(iii)]
of
T
.
Then
the
following
hold:
G
disc
disc
(i)
Let
α
∈
Aut
FC
(Π
disc
2
,
I)
,
J
∈
Cycle(Π
1
),
and
T
∈
Tpd
I
(Π
2/1
).
Then
it
holds
that
α
1
(J)
∈
Cycle(Π
disc
1
),
α(T
)
∈
Tpd
I
(Π
disc
2/1
).
G
disc
disc
Thus,
Aut
FC
(Π
disc
2
,
I)
acts
naturally
on
Cycle(Π
1
),
Tpd
I
(Π
2/1
).
G
(ii)
There
exists
a
unique
Aut
FC
(Π
disc
2
,
I)
-equivariant
[cf.
(i)]
map
disc
C
I
:
Cycle(Π
disc
1
)
−→
Tpd
I
(Π
2/1
)
such
that,
for
every
J
∈
Cycle(Π
disc
1
),
C
I
(J)
is
a
tripodal
sub-
G
group
associated
to
J.
Moreover,
for
every
α
∈
Aut
FC
(Π
disc
2
,
I)
∼
and
J
∈
Cycle(Π
disc
1
),
the
isomorphism
C
I
(J)
→
C
I
(α
1
(J))
induced
by
α
maps
every
lifting
cycle-subgroup
[cf.
Def-
inition
3.13,
(iii)]
of
C
I
(J)
bijectively
onto
a
lifting
cycle-
subgroup
of
C
I
(α
1
(J)).
(iii)
There
exists
an
assignment
Cycle(Π
disc
1
)
J
→
syn
I,J
—
where
syn
I,J
denotes
an
I-conjugacy
class
of
isomorphisms
∼
Π
tpd
→
C
I
(J)
—
such
that
(a)
syn
I,J
maps
I
tpd
bijectively
onto
I
in
a
fashion
that
is
∼
compatible
with
the
natural
isomorphism
I
tpd
→
I
in-
disc
disc
Π
disc
duced
by
the
projection
p
Π
{1,2,3}/{1,3}
:
Π
3
{1,3}
and
∼
disc
the
natural
outer
isomorphism
Π
disc
{1,3}
→
Π
{1,2}
obtained
by
switching
the
labels
“2”
and
“3”
[cf.
Corollary
2.17,
(ii);
Corollary
2.18,
(v);
[CbTpII],
Lemma
3.6,
(iv)],
∗
∗∗
,
J
tpd
bijectively
onto
lift-
(b)
syn
I,J
maps
the
subgroups
J
tpd
ing
cycle-subgroups
of
C
I
(J),
and
118
YUICHIRO
HOSHI
AND
SHINICHI
MOCHIZUKI
G
(c)
for
α
∈
Aut
FC
(Π
disc
2
,
I)
,
the
diagram
[of
I
tpd
-,
I-conjugacy
classes
of
isomorphisms]
Π
tpd
−−−→
⏐
⏐
syn
I,J
Π
tpd
⏐
⏐
syn
I,α
(J)
1
C
I
(J)
−−−→
C
I
(α
1
(J))
—
where
the
upper
horizontal
arrow
is
the
[uniquely
de-
termined
—
cf.
the
commensurable
terminality
of
I
tpd
of
Π
tpd
discussed
in
Corollary
2.18,
(v)]
I
tpd
-conjugacy
class
of
automorphisms
of
Π
tpd
that
lifts
T
Π
tpd
(α)
[cf.
Corol-
lary
2.20,
(v);
Theorem
2.24,
(iv)]
and
preserves
I
tpd
;
the
lower
horizontal
arrow
is
the
I-conjugacy
class
of
isomor-
phisms
induced
by
α
[cf.
(ii)]
—
commutes
up
to
possible
composition
with
the
cycle
symmetry
of
C
I
(α
1
(J))
as-
sociated
to
I
[cf.
Definition
3.13,
(iv)].
Finally,
the
assignment
J
→
syn
I,J
is
uniquely
determined,
up
to
possible
composition
with
cy-
cle
symmetries,
by
these
conditions
(a),
(b),
and
(c).
G
(iv)
Let
α
∈
Aut
FC
(Π
disc
2
,
I)
and
J
∈
Cycle(Π
1
).
Then
there
ex-
G
ists
an
automorphism
β
∈
Aut
FC
(Π
disc
2
,
I)
such
that
T
Π
tpd
(β)
[cf.
Corollary
2.20,
(v);
Theorem
2.24,
(iv)]
is
trivial,
and,
moreover,
α
1
(J)
=
β
1
(J).
Finally,
the
diagram
[of
I
tpd
-,
I-
conjugacy
classes
of
isomorphisms]
Π
tpd
⏐
⏐
syn
I,J
Π
tpd
⏐
⏐
syn
I,α
(J)
=syn
I,β
(J)
1
1
C
I
(J)
−−−→
C
I
(α
1
(J))
=
C
I
(β
1
(J))
—
where
the
lower
horizontal
arrow
is
the
isomorphism
induced
by
β
[cf.
(ii)]
—
commutes
up
to
possible
composition
with
the
cycle
symmetry
of
C
I
(α
1
(J))
=
C
I
(β
1
(J))
associated
to
I.
Proof.
Assertion
(i)
follows
from
the
various
definitions
involved.
As-
sertion
(ii)
follows
immediately
from
the
evident
discrete
version
[cf.
Corollaries
2.17,
(ii);
2.19,
(i)]
of
the
argument
involving
Remark
3.6.1
that
was
given
in
the
proof
of
Theorem
3.10,
(ii).
The
existence
portion
of
assertion
(iii)
follows,
in
light
of
Corollaries
2.17,
(ii);
2.20,
(i),
(v),
from
a
similar
argument
to
the
argument
applied
in
the
proof
of
the
ex-
istence
portion
of
Theorem
3.10,
(iii)
[cf.
also
the
fact
that
the
“syn
I,J
”
of
Theorem
3.10,
(iii),
was
constructed
from
a
suitable
geometric
outer
isomorphism].
The
uniqueness
portion
of
assertion
(iii)
follows
from
COMBINATORIAL
ANABELIAN
TOPICS
IV
119
the
compatibility
portion
of
condition
(a),
together
with
the
compu-
tation
of
discrete
outomorphism
groups
given
in
Theorem
2.24,
(ii).
Assertion
(iv)
follows
immediately
from
assertion
(iii),
together
with
a
similar
argument
to
the
argument
applied
in
the
proof
of
the
surjectiv-
ity
portion
of
Theorem
2.24,
(iv)
[cf.
the
argument
given
in
the
proof
of
Theorem
3.10,
(iv)].
This
completes
the
proof
of
Theorem
3.14.
Remark
3.14.1.
One
verifies
immediately
that
the
discrete
construc-
tions
of
Theorem
3.14,
(i),
(ii),
(iii),
(iv),
are
compatible,
in
an
evident
sense,
with
the
pro-Σ
constructions
of
Theorem
3.10,
(i),
(ii),
(iii),
(iv).
We
leave
the
routine
details
to
the
reader.
Remark
3.14.2.
One
verifies
immediately
that
remarks
analogous
to
Remarks
3.6.2,
3.10.1
in
the
profinite
case
may
be
made
in
the
dis-
crete
situation
treated
in
Theorem
3.14.
In
this
context,
we
observe
that
the
theory
of
the
“modules
of
local
orientations
Λ”
developed
in
[CbTpI],
§3,
admits
a
straightforward
discrete
analogue,
which
may
be
∼
applied
to
conclude
that
the
“orientation
isomorphisms
J
→
Λ
G
”
of
Remark
3.6.2,
(i),
are
compatible
with
the
natural
discrete
structures
on
the
domain
and
codomain.
Alternatively,
in
the
discrete
case,
relative
to
the
notation
of
Definition
2.2,
(iii),
one
may
think
of
these
modules
“Λ”
as
the
Z-duals
of
the
second
relative
singular
cohomology
modules
[with
Z-coefficients]
H
2
(U
X
,
∂U
X
;
Z)
—
cf.
the
discussion
of
orientations
in
[CbTpI],
Introduction.
Then
the
discrete
version
of
the
key
isomorphisms
[cf.
the
constructions
of
Remark
3.6.2]
of
[CbTpI],
Corollary
3.9,
(v),
(vi),
may
be
obtained
by
considering
the
connecting
homomorphism
[from
first
to
second
coho-
mology
modules]
in
the
long
exact
cohomology
sequence
associated
to
the
pair
(U
X
,
∂U
X
).
We
leave
the
routine
details
to
the
reader.
120
YUICHIRO
HOSHI
AND
SHINICHI
MOCHIZUKI
Appendix.
Explicit
limit
seminorms
associated
to
sequences
of
toric
surfaces
In
the
proof
of
Corollary
1.15,
(ii),
we
considered
sequences
of
dis-
crete
valuations
that
arose
from
vertices
or
edges
of
the
dual
semi-
graphs
associated
to
the
geometric
special
fibers
of
a
tower
of
coverings
of
stable
log
curves
and,
in
particular,
observed
that
the
convergence
of
a
suitable
subsequence
of
such
a
sequence
follows
immediately
from
the
general
theory
of
Berkovich
spaces.
In
the
present
Appendix,
we
reexamine
this
convergence
phenomenon
from
a
more
elementary
and
explicit
—
albeit
logically
unnecessary,
from
the
point
of
view
of
proving
Corollary
1.15,
(ii)!
—
point
of
view
that
only
requires
a
knowledge
of
elementary
facts
concerning
log
regular
log
schemes,
i.e.,
without
applying
the
terminology
and
notions
[e.g.,
of
“Stone-Čech
compact-
ifications”]
that
frequently
appear
in
the
general
theory
of
Berkovich
spaces
[cf.
the
proof
of
[Brk1],
Theorem
1.2.1].
In
particular,
we
discuss
the
notion
of
a
“stratum”
of
a
“toric
surface”
[cf.
Definition
A.1
below],
which
generalizes
the
notion
of
a
vertex
or
edge
of
the
dual
graph
of
the
special
fiber
of
a
stable
curve
over
a
complete
discrete
valuation
ring.
We
observe
that
such
a
stratum
determines
a
discrete
valuation
[cf.
Definition
A.4]
and
consider,
at
a
quite
explicit
level,
the
limit
of
a
suitable
subsequence
of
a
given
sequence
of
such
discrete
valuations
[cf.
Theorem
A.7
below].
The
material
presented
in
this
Appendix
is
quite
elementary
and
“well-known”,
but
we
chose
to
include
it
in
the
present
paper
since
we
were
unable
to
find
a
suitable
reference
that
discusses
this
material
from
a
similar
point
of
view.
In
the
present
Appendix,
let
R
be
a
complete
discrete
valuation
ring.
Write
K
for
the
field
of
fractions
of
R
and
S
log
for
the
log
scheme
def
obtained
by
equipping
S
=
Spec(R)
with
the
log
structure
determined
by
the
unique
closed
point
of
S.
Definition
A..1.
(i)
We
shall
refer
to
an
fs
log
scheme
X
log
over
S
log
as
a
toric
surface
over
S
log
if
the
following
conditions
are
satisfied:
(a)
The
underlying
scheme
X
of
X
log
is
of
finite
type,
flat,
and
of
pure
relative
dimension
one
[i.e.,
every
irreducible
component
of
every
fiber
of
the
underlying
morphism
of
schemes
X
→
S
is
of
dimension
one]
over
S.
(b)
The
fs
log
scheme
X
log
is
log
regular.
(c)
The
interior
[cf.,
e.g.,
[MT],
Definition
5.1,
(i)]
of
the
log
scheme
X
log
is
equal
to
the
open
subscheme
X
×
R
K
⊆
X
.
Given
two
toric
surfaces
over
S
log
,
there
is
an
evident
notion
of
isomorphism
of
toric
surfaces
over
S
log
.
(ii)
Let
X
log
be
a
toric
surface
over
S
log
[cf.
(i)]
and
n
a
nonnegative
integer.
Write
X
[n]
⊆
X
for
the
n-interior
of
X
log
[cf.
[MT],
Definition
5.1,
(i)]
and
X
[−1]
⊆
X
for
the
empty
subscheme.
COMBINATORIAL
ANABELIAN
TOPICS
IV
121
Then
we
shall
refer
to
a
connected
component
of
X
[n]
\
X
[n−1]
as
an
n-stratum
of
X
log
.
We
shall
write
Str
n
(X
log
)
for
the
set
of
n-strata
of
X
log
[so
Str
n
(X
log
)
=
∅
if
n
≥
3]
and
def
Str(X
log
)
=
Str
1
(X
log
)
Str
2
(X
log
).
Definition
A..2.
Let
I
be
a
totally
ordered
set
that
is
isomorphic
to
N
[equipped
with
its
usual
ordering].
In
particular,
it
makes
sense
to
speak
of
“limits
i
→
∞”
of
collections
of
objects
indexed
by
i
∈
I,
as
well
as
to
speak
of
the
“next
largest
element”
i
+
1
∈
I
associated
to
a
given
element
i
∈
I.
Then
we
shall
refer
to
a
sequence
of
fs
log
schemes
log
·
·
·
−−−→
X
i+1
−−−→
X
i
log
−−−→
·
·
·
—
where
i
ranges
over
the
elements
of
I
—
over
S
log
[indexed
by
I]
as
a
sequence
of
toric
surfaces
over
S
log
if,
for
each
i
∈
I,
X
i
log
is
a
toric
surface
over
S
log
[cf.
Definition
A.1,
(i)],
and,
moreover,
the
morphism
log
→
X
i
log
is
dominant.
Observe
that
the
horizontal
arrows
of
the
X
i+1
above
diagram
determine
[by
considering
the
induced
maps
of
generic
points
of
strata]
a
sequence
of
maps
of
sets
log
)
−→
Str(X
i
log
)
−→
·
·
·
.
·
·
·
−→
Str(X
i+1
Finally,
given
two
sequences
of
toric
surfaces
over
S
log
,
there
is
an
evident
notion
of
isomorphism
of
sequences
of
toric
surfaces
over
S
log
.
Definition
A..3.
Let
X
log
be
a
toric
surface
over
S
log
and
A
a
strict
henselization
of
X
at
[the
closed
point
determined
by]
z
∈
Str
2
(X
log
)
[cf.
Definition
A.1,
(i),
(ii)].
Write
F
for
the
field
of
fractions
of
A;
k
for
def
the
residue
field
of
A;
m
A
for
the
maximal
ideal
of
A;
X
z
=
Spec(A);
M
X
for
the
sheaf
of
monoids
on
X
that
defines
the
log
structure
of
X
log
;
M
for
the
fiber
of
M
X
/O
X
×
at
the
maximal
ideal
of
A;
def
def
def
Q
=
Hom(M,
Q
≥0
)
⊆
P
=
Hom(M,
R
≥0
)
⊆
V
=
Hom(M,
R)
—
where
we
write
Q
≥0
,
R
≥0
for
the
respective
submonoids
deter-
mined
by
the
nonnegative
elements
of
the
[additive
groups]
Q,
R
and
“Hom(M,
−)”
for
the
monoid
consisting
of
homomorphisms
of
monoids
from
M
to
“(−)”.
Thus,
one
verifies
easily
that
V
is
equipped
with
a
natural
structure
of
two-dimensional
vector
space
over
R.
In
the
fol-
lowing,
we
shall
use
the
superscript
“gp”
to
denote
the
groupification
of
any
of
the
monoids
of
the
above
discussion.
(i)
We
shall
say
that
a
submonoid
L
⊆
P
of
P
is
a
P
-ray
if
L
is
the
R
≥0
-orbit
of
some
nonzero
element
of
P
,
relative
to
the
natural
[multiplicative]
action
of
R
≥0
on
P
.
122
YUICHIRO
HOSHI
AND
SHINICHI
MOCHIZUKI
(ii)
We
shall
say
that
a
P
-ray
L
⊆
P
[cf.
(i)]
is
rational
(respec-
tively,
irrational)
if
L
∩
Q
=
{0}
(respectively,
L
∩
Q
=
{0}).
(iii)
Let
L
⊆
P
be
a
rational
P
-ray
[cf.
(i),
(ii)].
Then
we
shall
write
v
L
:
F
×
→
Q
⊆
R
for
the
discrete
valuation
associated
to
the
irreducible
component
of
the
blow-up
of
X
z
associated
to
L
⊆
P
,
normalized
so
as
to
map
each
prime
element
π
R
of
R
⊆
F
to
1
∈
Q.
That
is
to
say,
if
λ
∈
L
[which,
by
a
slight
abuse
of
notation,
we
regard
as
a
homomorphism
M
gp
→
R]
maps
π
R
→
1
∈
Q
[so
λ
∈
L
∩
Q],
and
f
∈
F
lies
in
the
A
×
-orbit
determined
by
m
∈
M
gp
,
then
v
L
(f
)
=
λ(m)
∈
Q.
Here,
we
observe
that
[one
verifies
easily
that]
the
submonoid
def
M
L
=
λ
−1
(Q
≥0
)
⊆
M
gp
is
isomorphic
to
Z
×
N.
In
particular,
if
we
denote
by
F
L
⊆
F
the
set
of
f
∈
F
that
lie
in
the
A
×
-orbits
determined
by
m
∈
M
L
and
write
A
L
⊆
F
for
the
A-subalgebra
generated
by
f
∈
F
L
,
then
the
“blow-up
of
X
z
associated
to
L”
referred
to
above
may
be
described
explicitly
as
def
X
L
=
Spec(A
L
)
−→
X
z
.
Indeed,
if
we
write
p
L
⊆
A
L
for
the
ideal
generated
by
the
set
of
f
∈
F
that
lie
in
the
A
×
-orbits
determined
by
the
noninvertible
elements
m
∈
M
L
,
then
it
follows
immediately
from
the
simple
structure
of
the
monoid
Z
×
N
that
p
L
is
the
prime
ideal
of
height
one
in
A
L
that
corresponds
to
the
discrete
valuation
v
L
,
and
that
the
k-algebra
A
L
/p
L
is
isomorphic
to
k[U,
U
−1
],
where
U
is
an
indeterminate.
(iv)
Write
M
S
for
the
sheaf
of
monoids
on
S
that
defines
the
log
structure
of
S
log
;
M
R
for
the
fiber
of
M
S
/O
S
×
at
the
unique
def
closed
point
of
S;
V
R
=
Hom(M
R
,
R).
Then
one
verifies
easily
that
V
R
is
a
one-dimensional
vector
space
over
R,
and
that
the
morphism
X
log
→
S
log
determines
an
R-linear
surjection
V
V
R
.
Let
e
α
,
e
β
∈
P
be
such
that
R
≥0
·
e
α
+
R
≥0
·
e
β
=
P
,
and,
moreover,
the
images
of
e
α
,
e
β
in
V
R
coincide.
[Note
that
the
existence
of
such
elements
e
α
,
e
β
∈
P
follows,
e.g.,
from
[ExtFam],
Proposition
1.7.]
Then
we
shall
refer
to
the
[necessarily
rational
—
cf.
(ii)]
P
-ray
R
≥0
·
(e
α
+
e
β
)
⊆
P
[cf.
(i)]
as
the
midpoint
P
-ray
at
z
∈
Str
2
(X
log
).
Here,
we
note
that
one
verifies
easily
that
the
P
-ray
R
≥0
·
(e
α
+
e
β
)
does
not
depend
on
the
choice
of
the
pair
(e
α
,
e
β
).
(v)
We
shall
refer
to
a
valuation
w
:
F
×
→
R
as
admissible
if
w
dominates
A
and
maps
each
prime
element
π
R
of
R
⊆
F
to
1
∈
R.
Let
w
be
an
admissible
valuation.
Then
by
restricting
w
to
the
elements
f
∈
F
that
lie
in
the
A
×
-orbits
determined
COMBINATORIAL
ANABELIAN
TOPICS
IV
123
by
m
∈
M
,
one
obtains
a
nonzero
homomorphism
of
monoids
M
→
R
≥0
,
i.e.,
an
element
of
P
.
We
shall
refer
to
the
P
-ray
L
w
determined
by
this
element
of
P
as
the
P
-ray
associated
to
the
admissible
valuation
w.
Thus,
if
L
w
is
rational
[cf.
(ii)],
then
it
follows
immediately
from
the
definitions
that,
in
the
notation
of
(iii),
the
valuation
of
A
determined
by
w
extends
to
a
valuation
of
A
L
w
(⊇
A).
Remark
A..3.1.
In
the
notation
of
Definition
A.3,
the
usual
topology
on
the
real
vector
space
V
naturally
determines
a
topology
on
the
subspace
P
⊆
V
,
as
well
as
on
the
set
of
P
-rays
[i.e.,
which
may
be
regarded
as
the
complement
of
the
“zero
element”
in
the
quotient
space
P/R
≥0
].
Moreover,
one
verifies
easily
that,
if
e
α
and
e
β
are
as
in
Definition
A.3,
(iv),
then
the
assignment
R
⊇
[0,
1]
γ
→
R
≥0
·
(γ
·
e
α
+
(1
−
γ)
·
e
β
)
determines
a
homeomorphism
of
the
closed
interval
[0,
1]
⊆
R
onto
the
resulting
topological
space
of
P
-rays,
and
that
the
subset
of
rational
P
-rays
is
dense
in
the
space
of
P
-rays.
In
particular,
it
makes
sense
to
speak
of
non-extremal
(respectively,
extremal)
P
-rays,
i.e.,
P
-rays
that
lie
(respectively,
do
not
lie)
in
the
interior
—
i.e.,
relative
to
the
homeomorphism
just
discussed,
the
open
interval
(0,
1)
⊆
[0,
1]
(respectively,
the
endpoints
{0,
1}
⊆
[0,
1])
—
of
the
space
of
P
-rays.
Finally,
we
observe
that
the
two
extremal
P
-rays
are
rational,
and
that
a
rational
P
-ray
is
non-extremal
if
and
only
if
its
associated
discrete
valuation
[cf.
Definition
A.3,
(iii)]
is
admissible
[cf.
Definition
A.3,
(v)].
Definition
A..4.
Let
X
log
be
a
toric
surface
over
S
log
;
z
∈
Str(X
log
)
[cf.
Definition
A.1,
(i),
(ii)].
Write
F
for
the
residue
field
of
the
generic
point
of
the
irreducible
component
of
X
on
which
[the
subset
of
X
determined
by]
z
∈
Str(X
log
)
lies.
Then
one
may
associate
to
z
∈
Str(X
log
)
a
collection
of
distinguished
valuations
on
F
,
as
well
as
a
uniquely
determined
canonical
valuation
on
F
,
as
follows:
(i)
If
z
is
a
1-stratum,
then
we
take
both
the
unique
distinguished
valuation
and
the
canonical
valuation
associated
to
z
to
be
the
discrete
valuation
F
×
−→
Q
⊆
R
associated
to
the
prime
of
height
1
determined
by
z,
normalized
so
as
to
map
each
prime
element
π
R
of
R
⊆
F
to
1
∈
Q.
(ii)
If
z
is
a
2-stratum,
then
we
take
the
collection
of
distinguished
valuations
associated
to
z
to
be
the
discrete
valuations
F
×
−→
Q
⊆
R
124
YUICHIRO
HOSHI
AND
SHINICHI
MOCHIZUKI
determined
by
the
restrictions
of
the
discrete
valuations
asso-
ciated
to
the
rational
P
-rays
[cf.
Definition
A.3,
(iii)].
We
take
the
canonical
valuation
associated
to
z
to
be
the
discrete
val-
uation
determined
by
the
restriction
of
the
discrete
valuation
associated
to
the
midpoint
P
-ray
at
z
[cf.
Definition
A.3,
(iii),
(iv)].
Here,
we
note
that
the
construction
from
z
of
either
the
collection
of
dis-
tinguished
valuations
or
the
uniquely
determined
canonical
valuation
is
functorial
with
respect
to
arbitrary
isomorphisms
of
pairs
(X
log
,
z)
[i.e.,
pairs
consisting
of
a
toric
surface
over
S
log
and
an
element
of
“Str(−)”
of
the
toric
surface].
Remark
A..4.1.
One
verifies
immediately
that
the
[noncuspidal]
val-
uations
of
the
discussion
preceding
Corollary
1.15
correspond
precisely
to
the
canonical
valuations
of
Definition
A.4.
Lemma
A..5
(Valuations
associated
to
irrational
rays).
In
the
notation
of
Definition
A.3,
let
L
⊆
P
be
an
irrational
P
-ray
[cf.
Definition
A.3,
(i),
(ii)],
{L
i
}
∞
i=1
a
sequence
of
P
-rays
such
that
L
=
lim
i→∞
L
i
[cf.
Remark
A..3.1],
and
{w
i
}
∞
i=1
a
sequence
of
admissible
valuations
such
that,
for
each
positive
integer
i,
L
i
is
the
P
-ray
asso-
ciated
to
w
i
[cf.
Definition
A.3,
(v)].
Then
there
exists
an
admissible
valuation
[cf.
Definition
A.3,
(v)]
v
L
:
F
×
−→
R
which
satisfies
the
following
conditions:
(a)
The
P
-ray
associated
to
v
L
[cf.
Definition
A.3,
(v)]
is
equal
to
L.
(b)
For
each
f
∈
F
×
,
it
holds
that
v
L
(f
)
=
lim
w
i
(f
).
i→∞
(c)
If
λ
∈
L
maps
a
prime
element
π
R
of
R
to
1
∈
R,
J
is
a
nonempty
finite
set,
{m
j
}
j∈J
is
a
collection
of
distinct
ele-
ments
of
M
gp
,
and
{f
j
}
j∈J
is
a
collection
of
elements
of
F
such
that
f
j
lies
in
the
A
×
-orbit
determined
by
m
j
,
then
v
L
(
f
j
)
=
min
λ(m
j
)
∈
R.
j∈J
j∈J
Moreover,
this
valuation
v
L
is
the
unique
admissible
valuation
[i.e.,
in
the
sense
of
Definition
A.3,
(v)]
that
satisfies
condition
(a).
In
partic-
ular,
v
L
depends
only
on
the
P
-ray
L
⊆
P
,
i.e.,
is
independent
of
∞
the
choice
of
the
sequences
{L
i
}
∞
i=1
and
{w
i
}
i=1
.
COMBINATORIAL
ANABELIAN
TOPICS
IV
125
Proof.
One
may
define
a
map
v
L
:
F
×
→
R
by
applying
the
formula
in
the
display
of
condition
(c)
in
the
case
where
J
=
1.
Then
one
verifies
easily
that
this
map
v
L
is
a
homomorphism
[with
respect
to
the
multiplicative
structure
of
F
×
]
and
satisfies
condition
(b).
Next,
let
us
observe
that
since
[we
have
assumed
that]
L
is
irrational,
the
map
M
gp
→
R
determined
by
λ
∈
L
is
injective.
Thus,
it
follows
from
condition
(b),
together
with
the
fact
that
each
of
the
w
i
’s
is
a
valuation,
that
the
map
v
L
satisfies
condition
(c),
which
implies
that
the
map
v
L
is
a
[necessarily
admissible]
valuation
on
F
.
Moreover,
it
follows
immediately
from
the
definition
of
v
L
that
v
L
satisfies
condition
(a).
This
completes
the
proof
of
Lemma
A.5.
Lemma
A..6
(Convergence
of
midpoints
of
closed
intervals).
Let
def
·
·
·
⊆
[a
i+1
,
b
i+1
]
⊆
[a
i
,
b
i
]
⊆
[a
i−1
,
b
i−1
]
⊆
·
·
·
⊆
[a
0
,
b
0
]
=
[0,
1]
⊆
R
—
where
i
ranges
over
the
nonnegative
integers
—
be
a
sequence
of
inclusions
of
nonempty
closed
intervals
in
[0,
1].
For
each
i,
write
c
i
for
def
the
midpoint
of
the
closed
interval
[a
i
,
b
i
],
i.e.,
c
i
=
(a
i
+b
i
)/2
∈
[a
i
,
b
i
].
Then
the
sequence
of
midpoints
{c
i
}
∞
i=1
converges.
Proof.
This
follows
immediately
from
the
[easily
verified]
fact
that
the
∞
sequences
{a
i
}
∞
i=1
,
{b
i
}
i=1
converge.
Theorem
A..7
(Explicit
limit
seminorms
associated
to
sequences
of
toric
surfaces).
Let
R
be
a
complete
discrete
valuation
ring
and
I
a
totally
ordered
set
that
is
isomorphic
to
N
[equipped
with
its
usual
ordering].
Write
K
for
the
field
of
fractions
of
R
and
S
log
for
the
def
log
scheme
obtained
by
equipping
S
=
Spec(R)
with
the
log
structure
determined
by
the
unique
closed
point
of
S.
Let
log
·
·
·
−−−→
X
i+1
−−−→
X
i
log
−−−→
·
·
·
be
a
sequence
of
toric
surfaces
over
S
log
indexed
by
I
[cf.
Definition
A.2]
and
{z
i
}
i∈I
∈
lim
Str(X
i
log
)
←−
i∈I
[cf.
Definitions
A.1,
(ii);
A.2].
Then,
after
possibly
replacing
I
by
a
suitable
cofinal
subset
of
I,
there
exist
sequences
{v
i
:
F
i
×
→
R}
i∈I
,
{v
z
i
}
i∈I
—
where,
for
each
i
∈
I,
F
i
denotes
the
residue
field
of
some
point
x
i
∈
X
i
×
R
K;
v
i
:
F
i
×
→
R
is
a
valuation;
v
z
i
is
a
distinguished
valuation
associated
to
z
i
[cf.
Definition
A.4]
—
such
that
126
YUICHIRO
HOSHI
AND
SHINICHI
MOCHIZUKI
(a)
v
i
maps
each
prime
element
of
R
⊆
F
i
to
1
∈
R
[which
thus
implies
that
v
i
dominates
R];
(b)
the
x
i
’s
and
v
i
’s
are
compatible
[in
the
evident
sense]
with
log
→
X
i
log
of
the
above
respect
to
the
upper
horizontal
arrows
X
i+1
diagram;
(c)
for
every
nonzero
rational
function
f
on
the
irreducible
compo-
nent
of
X
i
containing
x
i
that
is
regular
at
x
i
,
hence
determines
an
element
f
∈
F
i
[cf.
Remark
A..7.1
below],
it
holds
that
v
i
(f
)
=
lim
v
z
j
(f
)
j→∞
[cf.
Definition
A.4]
—
where
j
ranges
over
the
elements
of
I
that
are
≥
i,
and
we
regard
v
i
as
a
map
defined
on
F
i
by
sending
F
i
0
→
+∞.
Finally,
these
sequences
of
valuations
{v
i
}
i∈I
,
{v
z
i
}
i∈I
may
be
con-
structed
in
a
way
that
is
functorial
[in
the
evident
sense]
with
respect
to
isomorphisms
of
pairs
consisting
of
a
sequence
of
toric
surfaces
over
S
log
and
a
compatible
collection
of
strata
[i.e.,
“{z
i
}
i∈I
”].
Proof.
Until
further
notice,
we
take,
for
each
i
∈
I,
v
z
i
to
be
the
canon-
ical
valuation
associated
to
z
i
[cf.
Definition
A.4].
Next,
let
us
observe
that
one
verifies
easily
that
we
may
assume
without
loss
of
generality,
by
replacing
I
by
a
suitable
cofinal
subset
of
I,
that
there
exists
an
element
n
∈
{1,
2}
such
that
every
member
of
{z
i
}
is
an
n-stratum,
i.e.,
one
of
the
following
conditions
is
satisfied:
(1)
Every
member
of
{z
i
}
is
a
1-stratum.
(2)
Every
member
of
{z
i
}
is
a
2-stratum.
First,
we
consider
Theorem
A.7
in
the
case
where
condition
(1)
is
satisfied.
For
each
i
∈
I,
write
Z
i
⊆
X
i
for
the
reduced
closed
sub-
scheme
of
X
i
whose
underlying
closed
subset
[⊆
X
i
]
is
the
closure
of
the
subset
of
X
determined
by
the
1-stratum
z
i
.
Then
let
us
observe
that
if,
after
possibly
replacing
I
by
a
suitable
cofinal
subset
of
I,
it
holds
that,
for
each
i
∈
I,
the
composite
Z
i+1
→
X
i+1
→
X
i
is
quasi-finite,
then
the
system
consisting
of
the
v
z
i
’s
[cf.
Definition
A.4,
(i)]
already
yields
a
system
of
valuations
{v
i
}
i∈I
as
desired.
Thus,
we
may
assume
without
loss
of
generality,
by
replacing
I
by
a
suitable
cofinal
subset
of
I,
that,
for
each
i
∈
I,
the
composite
Z
i+1
→
X
i+1
→
X
i
is
not
quasi-
finite,
i.e.,
that
the
image
of
this
composite
is
a
closed
point
y
i
∈
X
i
of
X
i
.
Here,
we
observe
that
since
we
are
operating
under
the
assumption
that
condition
(1)
is
satisfied,
it
follows
from
the
fact
that
z
i+1
→
z
i
that
y
i
necessarily
lies
in
the
regular
locus
of
X
i
.
For
each
i
∈
I,
write
B
i
for
the
local
ring
of
X
i
at
y
i
∈
X
i
,
E
i
for
the
field
of
fractions
of
B
i
,
and
v
z
i
:
E
i
×
→
R
for
the
discrete
valuation
defined
in
Definition
A.4,
(i).
Thus,
one
verifies
immediately
that
the
morphisms
·
·
·
→
X
i+1
→
X
i
→
·
·
·
COMBINATORIAL
ANABELIAN
TOPICS
IV
127
induce
compatible
chains
of
injections
·
·
·
→
B
i
→
B
i+1
→
·
·
·
,
·
·
·
→
E
i
→
E
i+1
→
·
·
·
.
Moreover,
if
π
R
is
a
prime
element
of
R,
then
the
discrete
valuation
v
z
i
may
be
interpreted
as
the
discrete
valuation
of
B
i
determined
by
the
unique
height
one
prime
of
B
i
that
contains
π
R
.
In
particular,
since
B
i
is
regular,
hence
a
unique
factorization
domain,
one
verifies
immediately
—
by
considering
the
extent
to
which
positive
powers
of
an
element
f
∈
B
i
are
divisible,
in
B
i
or
in
B
i+1
,
by
positive
powers
of
π
R
—
that,
for
each
i
∈
I
and
f
∈
B
i
,
it
holds
that
(0
≤)
v
z
i
(f
)
≤
v
z
i+1
(f
).
(∗)
For
each
i
∈
I,
write
def
p
i
=
{
f
∈
B
i
|
lim
v
z
j
(f
)
=
+∞
}
⊆
B
i
.
j→∞
Then
since
each
v
z
j
is
a
[discrete]
valuation,
one
verifies
immediately
that
p
i
⊆
B
i
is
a
prime
ideal
of
B
i
.
Moreover,
since
π
R
∈
p
i
,
we
conclude
that
the
ideal
p
i
is
not
maximal,
i.e.,
that
the
height
of
p
i
is
∈
{0,
1}.
Next,
let
us
observe
that
if,
after
possibly
replacing
I
by
a
suitable
cofinal
subset
of
I,
it
holds
that,
for
each
i
∈
I,
the
prime
ideal
p
i
is
of
height
1,
then
it
follows
immediately
that
p
i
determines
a
closed
point
x
i
of
the
generic
fiber
of
X
i
,
and
that,
if
we
write
F
i
for
the
residue
field
of
X
i
at
x
i
and
v
i
:
F
i
×
→
R
for
the
uniquely
determined
[since
F
i
is
a
finite
extension
of
K]
discrete
valuation
on
F
i
that
extends
the
given
discrete
valuation
on
K
and
maps
π
R
→
1
∈
R,
then
the
limit
lim
j→∞
v
z
j
(−)
[cf.
(∗)]
determines
a
valuation
on
F
i
=
(B
i
)
p
i
/p
i
(B
i
)
p
i
that
necessarily
coincides
[since
F
i
is
a
finite
extension
of
K]
with
v
i
;
in
particular,
one
obtains
a
system
of
valuations
{v
i
}
i∈I
as
desired.
Thus,
we
may
assume
without
loss
of
generality,
by
replacing
I
by
a
suitable
cofinal
subset
of
I,
that,
for
each
i
∈
I,
the
prime
ideal
p
i
is
of
height
0,
i.e.,
p
i
=
{0},
hence
determines
a
generic
point
x
i
of
some
irreducible
component
of
X
i
such
that
E
i
may
be
naturally
identified
with
the
residue
field
F
i
of
X
i
at
x
i
.
But
this
implies
that,
for
f
∈
E
i
×
=
F
i
×
,
the
quantity
def
v
i
(f
)
=
lim
v
z
j
(f
)
∈
R
j→∞
is
well-defined
[cf.
(∗)].
Moreover,
one
verifies
immediately
that
this
definition
of
v
i
determines
a
valuation
on
E
i
=
F
i
.
In
particular,
one
obtains
a
system
of
valuations
{v
i
}
i∈I
as
desired.
This
completes
the
proof
of
Theorem
A.7
in
the
case
where
condition
(1)
is
satisfied.
Next,
we
consider
Theorem
A.7
in
the
case
where
condition
(2)
is
satisfied.
For
each
i
∈
I,
write
Q
i
,
P
i
,
V
i
for
the
objects
“Q”,
“P
”,
“V
”
defined
in
Definition
A.3
in
the
case
where
we
take
the
data
“(X
log
,
z
∈
Str
2
(X
log
))”
in
Definition
A.3
to
be
(X
i
log
,
z
i
∈
Str
2
(X
i
log
)).
Then
one
128
YUICHIRO
HOSHI
AND
SHINICHI
MOCHIZUKI
log
verifies
easily
that
the
morphism
X
i+1
→
X
i
log
determines
a
nontrivial
R-linear
map
V
i+1
→
V
i
that
maps
Q
i+1
,
P
i+1
⊆
V
i+1
into
Q
i
,
P
i
⊆
V
i
,
respectively.
Next,
let
us
observe
that
if,
after
possibly
replacing
I
by
a
suitable
cofinal
subset
of
I,
it
holds
that,
for
each
i
∈
I,
the
R-linear
map
V
i+1
→
V
i
is
of
rank
one,
i.e.,
the
image
of
P
i+1
⊆
V
i+1
in
V
i
is
a
rational
P
i
-ray
L
i
[cf.
Definition
A.3,
(i),
(ii)],
then
we
may
assume
without
loss
of
generality,
by
taking
v
z
i
to
be
the
distinguished
valuation
associated
to
the
rational
P
i
-ray
L
i
[cf.
Definition
A.4,
(ii);
Remark
A..7.2
below]
and
then
replacing
the
pair
(X
i
,
z
i
)
by
the
pair
consisting
of
the
blow-
up
of
X
i
and
the
1-stratum
of
this
blow-up
determined
by
L
i
[cf.
the
discussion
of
Definition
A.3,
(iii)],
that
condition
(1)
is
satisfied.
Thus,
we
may
assume
without
loss
of
generality,
by
replacing
I
by
a
suitable
cofinal
subset
of
I,
that,
for
each
i
∈
I,
the
R-linear
map
V
i+1
→
V
i
is
of
rank
=
1,
hence
[cf.
the
existence
of
the
R-linear
surjection
“V
V
R
”
of
Definition
A.3,
(iv)]
of
rank
two,
i.e.,
an
isomorphism.
Since
the
R-linear
map
V
i+1
→
V
i
is
an
isomorphism,
it
follows
im-
mediately
from
Lemma
A.6,
together
with
Remark
A..3.1,
that,
for
each
i
∈
I,
the
sequence
consisting
of
the
images
in
P
i
of
the
midpoint
P
j
-
rays
[cf.
Definition
A.3,
(iv)],
where
j
ranges
over
the
elements
of
I
such
that
j
≥
i,
converges
to
a
[not
necessarily
rational]
P
i
-ray
L
i,∞
⊆
P
i
.
If,
after
possibly
replacing
I
by
a
suitable
cofinal
subset
of
I,
it
holds
that,
for
each
i
∈
I,
the
P
i
-ray
L
i,∞
is
rational,
then
we
may
assume
without
loss
of
generality,
by
taking
v
z
i
to
be
the
distinguished
valua-
tion
associated
to
the
rational
P
i
-ray
L
i,∞
[cf.
Definition
A.4,
(ii);
Remark
A..7.2
below]
and
then
replacing
the
pair
(X
i
,
z
i
)
by
the
pair
consisting
of
the
blow-up
of
X
i
and
the
1-stratum
of
this
blow-up
deter-
mined
by
L
i,∞
[cf.
the
discussion
of
Definition
A.3,
(iii)],
that
condition
(1)
is
satisfied.
Thus,
it
remains
to
consider
the
case
in
which
we
may
assume
without
loss
of
generality,
by
replacing
I
by
a
suitable
cofinal
subset
of
I,
that,
for
each
i
∈
I,
the
P
i
-ray
L
i,∞
is
irrational.
Then
the
system
consisting
of
the
valuations
v
L
i,∞
’s
of
Lemma
A.5
yields
a
system
of
valuations
{v
i
}
i∈I
as
desired.
This
completes
the
proof
of
Theorem
A.7.
Remark
A..7.1.
In
the
situation
of
Theorem
A.7,
for
I
j
≥
i,
write
z
j
i
for
the
irreducible
locally
closed
subset
of
X
i
determined
by
the
image
of
the
stratum
z
j
in
X
i
.
Thus,
z
j
i
⊆
z
j
i
for
all
j
≥
j,
and
one
verifies
immediately
that
the
intersection
def
i
z
∞
=
z
j
i
j≥i
is
nonempty.
Moreover,
it
follows
immediately
from
the
constructions
i
discussed
in
the
proof
of
Theorem
A.7
that
if
ξ
i
∈
z
∞
,
then
any
element
COMBINATORIAL
ANABELIAN
TOPICS
IV
129
f
of
the
local
ring
O
X
i
,ξ
i
of
X
i
at
ξ
i
determines
a
rational
function
on
the
irreducible
component
of
X
i
containing
x
i
that
is
regular
at
x
i
[cf.
Theorem
A.7,
(c)].
Remark
A..7.2.
Although,
in
certain
cases
[cf.
the
final
portion
of
the
proof
of
Theorem
A.7],
the
distinguished
valuation
v
z
i
in
the
statement
of
Theorem
A.7
is
not
necessarily
canonical,
the
system
of
valuations
{v
i
}
i∈I
obtained
in
Theorem
A.7
is
nevertheless
sufficient
[cf.
the
func-
toriality
discussed
in
the
final
portion
of
Theorem
A.7]
to
derive
the
conclusion
of
Corollary
1.15,
(ii),
i.e.,
without
applying
the
theory
of
[Brk1].
130
YUICHIRO
HOSHI
AND
SHINICHI
MOCHIZUKI
References
[André]
Y.
André,
On
a
geometric
description
of
Gal(Q
p
/Q
p
)
and
a
p-adic
avatar
Duke
Math.
J.
119
(2003),
1-39.
of
GT,
[Brk1]
V.
G.
Berkovich,
Spectral
theory
and
analytic
geometry
over
non-
Archimedean
fields,
Mathematical
Surveys
and
Monographs,
33.
American
Mathematical
Society,
Providence,
RI,
1990.
[Brk2]
V.
G.
Berkovich,
Smooth
p-adic
analytic
spaces
are
locally
contractible,
Invent.
Math.
137
(1999),
1-84.
[Bgg1]
M.
Boggi,
Profinite
Teichmüller
theory,
Math.
Nachr.
279
(2006),
953-987.
[Bgg2]
M.
Boggi,
The
congruence
subgroup
property
for
the
hyperelliptic
modular
group,
preprint
(arXiv:math.0803.3841v4
18
Jan
2013).
[Bgg3]
M.
Boggi,
On
the
procongruence
completion
of
the
Teichmüller
modular
group,
Trans.
Amer.
Math.
Soc.
366
(2014),
5185-5221.
[DM]
P.
Deligne
and
D.
Mumford,
The
irreducibility
of
the
space
of
curves
of
given
genus,
Inst.
Hautes
Études
Sci.
Publ.
Math.
36
(1969),
75-109.
[NodNon]
Y.
Hoshi
and
S.
Mochizuki,
On
the
combinatorial
anabelian
geometry
of
nodally
nondegenerate
outer
representations,
Hiroshima
Math.
J.
41
(2011),
275-342.
[CbTpI]
Y.
Hoshi
and
S.
Mochizuki,
Topics
surrounding
the
combinatorial
an-
abelian
geometry
of
hyperbolic
curves
I:
Inertia
groups
and
profinite
Dehn
twists,
Galois-Teichmüller
Theory
and
Arithmetic
Geometry,
659-811,
Adv.
Stud.
Pure
Math.,
63,
Math.
Soc.
Japan,
Tokyo,
2012.
[CbTpII]
Y.
Hoshi
and
S.
Mochizuki,
Topics
surrounding
the
combinatorial
an-
abelian
geometry
of
hyperbolic
curves
II:
Tripods
and
combinatorial
cuspidal-
ization,
RIMS
Preprint
1762
(November
2012).
[CbTpIII]
Y.
Hoshi
and
S.
Mochizuki,
Topics
surrounding
the
combinatorial
an-
abelian
geometry
of
hyperbolic
curves
III:
Tripods
and
tempered
fundamental
groups,
RIMS
Preprint
1763
(November
2012).
[KN]
K.
Kato
and
C.
Nakayama,
Log
Betti
cohomology,
log
étale
cohomology,
and
log
de
Rham
cohomology
of
log
schemes
over
C,
Kodai
Math.
J.
22
(1999),
161-186.
[LocAn]
S.
Mochizuki,
The
local
pro-p
anabelian
geometry
of
curves,
Invent.
Math.
138
(1999),
319-423.
[ExtFam]
S.
Mochizuki,
Extending
families
of
curves
over
log
regular
schemes,
J.
Reine
Angew.
Math.
511
(1999),
43-71.
[CanLift]
S.
Mochizuki,
The
absolute
anabelian
geometry
of
canonical
curves,
Kazuya
Kato’s
fiftieth
birthday.
Doc.
Math.
2003,
Extra
Vol.,
609-640.
[GeoAn]
S.
Mochizuki,
The
geometry
of
anabelioids,
Publ.
Res.
Inst.
Math.
Sci.
40
(2004),
819-881.
[SemiAn]
S.
Mochizuki,
Semi-graphs
of
anabelioids,
Publ.
Res.
Inst.
Math.
Sci.
42
(2006),
221-322.
[CmbGC]
S.
Mochizuki,
A
combinatorial
version
of
the
Grothendieck
conjecture,
Tohoku
Math.
J.
(2)
59
(2007),
455-479.
[AbsCsp]
S.
Mochizuki,
Absolute
anabelian
cuspidalizations
of
proper
hyperbolic
curves,
J.
Math.
Kyoto
Univ.
47
(2007),
451-539.
[AbsTpII]
S.
Mochizuki,
Topics
in
Absolute
Anabelian
Geometry
II:
Decomposition
Groups
and
Endomorphisms,
J.
Math.
Sci.
Univ.
Tokyo
20
(2013),
171-269.
[CmbCsp]
S.
Mochizuki,
On
the
combinatorial
cuspidalization
of
hyperbolic
curves,
Osaka
J.
Math.
47
(2010),
651-715.
[IUTeichI]
S.
Mochizuki,
Inter-universal
Teichmüller
Theory
I:
Construction
of
Hodge
Theaters,
to
appear
in
Publ.
Res.
Inst.
Math.
Sci.
COMBINATORIAL
ANABELIAN
TOPICS
IV
131
[MT]
S.
Mochizuki
and
A.
Tamagawa,
The
algebraic
and
anabelian
geometry
of
configuration
spaces,
Hokkaido
Math.
J.
37
(2008),
75-131.
[NO]
C.
Nakayama
and
A.
Ogus,
Relative
rounding
in
toric
and
logarithmic
geom-
etry,
Geom.
Topol.
14
(2010),
2189-2241.
[Prs]
L.
Paris,
Residual
p
properties
of
mapping
class
groups
and
surface
groups,
Trans.
Amer.
Math.
Soc.
361
(2009),
2487-2507.
[PS]
F.
Pop
and
J.
Stix,
Arithmetic
in
the
fundamental
group
of
a
p-adic
curve.
On
the
p-adic
section
conjecture
for
curves,
J.
Reine
Angew.
Math.
725
(2017),
1-40.
[RZ]
L.
Ribes
and
P.
Zalesskii,
Profinite
groups,
Second
edition.
Ergebnisse
der
Mathematik
und
ihrer
Grenzgebiete.
3.
Folge.
A
Series
of
Modern
Surveys
in
Mathematics,
40.
Springer-Verlag,
Berlin,
2010.
[Tama1]
A.
Tamagawa,
The
Grothendieck
conjecture
for
affine
curves,
Compositio
Math.
109
(1997),
135-194.
[Tama2]
A.
Tamagawa,
Resolution
of
nonsingularities
of
families
of
curves,
Publ.
Res.
Inst.
Math.
Sci.
40
(2004),
1291-1336.
[ZM]
P.
A.
Zalesskii
and
O.
V.
Mel’nikov,
Subgroups
of
profinite
groups
acting
on
trees,
Math.
USSR-Sb.
63
(1989),
405-424.
(Yuichiro
Hoshi)
Research
Institute
for
Mathematical
Sciences,
Ky-
oto
University,
Kyoto
606-8502,
JAPAN
Email
address:
yuichiro@kurims.kyoto-u.ac.jp
(Shinichi
Mochizuki)
Research
Institute
for
Mathematical
Sciences,
Kyoto
University,
Kyoto
606-8502,
JAPAN
Email
address:
motizuki@kurims.kyoto-u.ac.jp